THE  LIBRARY 

OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

LOS  ANGELES 

GIFT  OF 

Jolm  S.Prell 


^^  L 


AN 


ELEMENTARY  TREATISE 


ON- 


MECHANICS. 


Br  I.  W.  JACKSON, 

PEOFESSOR   OF   MATHEMATICS   IN   UNION   COLLEGE. 

JOHIV  S.  l»r^ELL 

Civil  &  Mechanical  Engineer* 

SAN  FRANCISCO,  OAL. 

FOURTH  EDITION. 


SCHENECTADY: 
1870. 


Entered  according  to  act  of  Congress,  in  the  oflace  of  the  Clerk  of  the  Xorthei-n 
District  of  New  York. 


CHAELES  VAN  BENTHUTSEN  AND  SONS, 

Printers,  Stereotypers  and  Binders,  Albany. 


J/3 


£^ 


ADYEETISEMENT. 


The  following  exposition  of  the  elementary  principles  of  rational 
mechanics,  is  intended  to  be  sufficiently  brief  and  free  from  diffi- 
culties to  be  thoroughly  mastered  by  students  of  ordinary  capacity, 
in  the  time  usually  allotted  to  the  subject,  and  yet  comprehensive 
enough  to  be  made  the  basis  of  a  course  of  general  physics  and 
practical  mechanics.  It  has  no  claims  to  originality  ;  the  methods 
employed  being  generally  those  which  have  been  long  in  use,  and 
which  may  be  found  in  many  of  the  best  treatises  on  the  same 
subject. 

Of  the  works  consulted  in  the  preparation  of  this  volume,  most 
assistance  has  been  derived  from  those  of  Poisson  and  Bou- 
CHARLAT ;  occasional  use  has  also  been  made  of  those  of  Earn- 
SHAW,  Potter, Sauri, Puissant, Franc(EUR,  and  D'Aubuisson. 


737386 

KnRinecring 
Library 


TABLE  OF  CONTENTS, 


INTRODUCTION. 


Definitions,        --_--__-__  9 

Laws  of  motion      -------__  10 

PART   FIRST. 

STATICS. 

Definitions,  -----_--__     14 

Parallelogram  of  forces,      -         -        -        -        -        -        -        -        18 

Methods  of  determining  the  position  of  a  point,       -        -        -        -     21 

Of  forces  situated  in  the  same  plane,  and  applied  at  the  same  point,     26 
Of  forces  applied  at  the  same  point,  and  situated  in  different  planes,     30 
Of  parallel  forces,  ---------33 

Of  oblique  forces,  applied  at  different  points,  and  situated  in  the 
same  plane,  --_______        45 

Of  forces  applied  at  different  points,  and  situated  in  different  planes,     56 
Centre  of  gravity,      --------        -QQ 

Machines,      -----------77 

The  rope  machine,       ---_---__        73 
The  lever,      ---  _______     gQ 

The  inclined  plane,     _-_---___82 

The  pulley,  --  84 

The  wheel  and  axle,  _-__--__        37 

The  screw,  __________     39 

The  wedge,     .__-___-___        92 
General  principle  of  equilibrium  in  machines,  -         -         -         -     94 

Friction,     ----,-----__        98 


Vi  CONTENTS. 

PART    SECOND. 
DYNAMICS. 

Of  the  rectilinear  motion  of  a  material  point,           _        _        _         -  IQl 

Of  the  motion  of  bodies  upon  inclined  planes,     -        -        -        -  111 

Of  the  motion  of  a  material  point  on  a  given  curve,         -        -         -  114 

Of  the  simple  pendulum,             ____---  118 

Of  central  forces,             ____-----  124 

Of  projectiles,           ___-                  __._  133 
Measure  of  forces,          -        -        -        -         -        -         -         -         -139 

Principle  of  D'Alembert,            ____---  145 

Moment  of  inertia,          __-_-_---  148 

The  compound  pendulum,           __-_-__  156 

Of  the  collision  of  bodies,       --------  164 

Gravitation,      --------.-  178 

PART    THIRD. 
HYDROSTATICS. 

Definitions  and  fundamental  principle,        -----         185 

Pressure  due  to  the  gravity  of  the  fluid,  -----  188 

Principle  of  virtual  velocities  observed,       -----        188 

Compressible  fluids,        ---------  190 

The  equiUbrium  of  fluids,  regard  being  had  to  their  gravity        -        191 
Pressure  exerted  by  a  fluid  upon  the  bottom  of  the  vessel  contain- 
ing it,  ----------         191 

Equilibrium  in  the  case  of  connected  vessels,  _         _         _         -  195 

Pressure  of  a  fluid  upon  the  surface  of  the  containing  vessel,  or  on 
any  immersed  surface,  -         -         -         -         -        -         -         196 

Centre  of  pressure,         _-,-------  198 

Applications  of  the  formulae  for  the  centre  of  pressure,       -        -        201 
Of  the  resultant  of  the  pressures  exerted  by  a  fluid  upon  the  surface 
of  a  body  either  wholly  or  partially  immersed,     -        -        -        -  205 

Conditions  of  equilibrium  of  an  immersed  solid,         -         -         -         210 
The  resultant  of  the  pressures  exerted  by  a  fluid  upon  the  whole 
interior  surface  of  the  vessel  containing  it,  -        -        -        -  211 

Of  the  equilibrium  of  floating  bodies,         -----        212 

Stability  of  floating  bodies,  -------  220 

Specific  gravity,         ----         ___._        222 


CONTENTS.  VU 

Hydrometers,  __-------        224 

Pressure  of  the  atmosphere,  elasticity  of  compressible  fluids  and  the 
barometer,  ----------  227 

Measurement  of  heights  by  the  barometer,        -        -        -        -        232 

PART    FOURTH. 
HYDRODYNAMICS. 

Efflux  of  fluids, 239 

Discharge ;  theoretic  discharge ;  actual  discharge,     -        -        -        -  242 

Contraction  of  the  fluid  vein,     -------        242 

Efflux  through  short  tubes,  __._---  243 

Efflux  through  large  rectangular  lateral  orifices,  _        -        -        244 

Efflux  under  variable  pressure,       -------  246 

INSTRUMENTS  AND  MACHINES. 

The  Siphon,     ----------        249 

The  sucking  pump,         ..--.-----  250 

The  forcing  pump,  _._-__--        258 

Bramah's  press,    ----------  258 

The  air  pump,  --         -------        259 


ELEMENTS  OE  MECHANICS 


INTRODUCTION. 


1.  When  a  body  occupies  successively  different  jiosi- 
tions  in  space,  it  is  said  to  be  in  motion. 

2.  Whatever  produces  or  tends  to  produce  motion, 
is  called  force.  The  action  of  a  force,  whatever  its 
origin,  may  be  conceived  to  consist  in  communicating 
to  the  body  on  which  it  acts,  impulses,  either  finite, 
or  infinitely  small.  The  body  is,  in  all  cases,  supposed 
to  be  entirely  inert,  and  subject  only  to  the  influence 
of  forces  exterior  to  itself."^ 

3.  It  can  readily  be  conceived  that  two  or  more 
forces  may  be  so  applied  to  a  body,  that  their  eflects 
shall  counteract  each  other,  and  no  motion  shall 
ensue :  in  this  case,  the  forces  are  said  to  be  in  equi- 
lihri%m. 

*  Abstraction  is  thus  made  of  vitality,  and  certain  properties  inherent 
in  matter,  as  gravity  and  the  chemical  and  electrical  attractions,  etc.  etc. 
When  the  effects  of  these  are  to  be  determined,  they  are  regarded  as 
extraneous  forces. 

2 


10  MECHANICS. 

4.  Mechanics  is  the  science  which  treats  of  equi- 
librium  and  of  motion.  It  is  divided  into  two  parts, 
Statics  and  Dynamics. 

5.  In  statics,  the  subject  of  equilibrium  is  consi- 
dered. It  is  divided  into  two  parts  : 

I.  The  statics  of  solids,  called  simply  Statics ; 

II.  The  statics  of  fluids,  called  Hydrostatics. 

6.  In  dynamics,  the  subject  of  motion  is  considered. 
It  also  is  divided  into  two  parts  : 

I.  The  dynamics  of  solids,  called  ^ixR-^ly  Dynamics; 

II.  The  dynamics  of  fluids,  or  Hydrodynamics. 

7.  The  reasonings  in  mechanics  are  based  upon 
certain  facts  of  great  generality,  derived  from  ob- 
servation and  experiment,  called  the  laws  of  motion. 
We  shall  do  little  more  than  enunciate  them,  leaving 
it  chiefly  to  the  instructor  to  supply  the  observations 
and  experiments  by  Avhich  they  are  proved  and  il- 
lustrated. 

laws  of  motion. 

I.   The  law  of  inertia. 

The  motion  of  a  body,  when  left  to  itself  is  rectilinear 
and  uniform.  Thus  when  a  body  receives  an  impulse 
and  then  is  abandoned  to  itself,  according  to  this 
law  all  its  points  will  describe  straight  lines,  and 
will  move  over  equal  spaces  in  equal  times. 

At  flrst  view,  this  law  appears  to  be  contrary  to 
the  most  obvious  facts ;  for  the  motions  wdiich  we 
observe  in  bodies  on  the  surface  of  the  earth  are 
neither  rectilinear  nor  uniform,  and  they  all  soon 


INTRODUCTION.  11 

terminate.  In  these  cases,  however,  the  bodies  are 
really  not  left  to  themselves,  but  are  constantly 
acted  upon  by  certain  forces,  as  gravity,  friction,  etc., 
which  in  a  greater  or  less  degree  interfere  with  their 
motions;  but  it  is  invariably  found,  that  as  these 
interfering  forces  are  diminished  in  intensity,  the 
motions  take  place  more  nearly  in  accordance  with 
the  law.  Thus  in  the  case  of  a  body  moving  on  a 
horizontal  plane,  as  the  surface  of  each  is  rendered 
smoother,  the  more  nearly  does  the  motion  become 
rectilinear,  and  the  longer  does  it  continue.  Again,  a 
pendulum,  oscillating  under  ordinary  circumstances, 
is  soon  brought  to  rest ;  but  if  we  diminish  the  fric- 
tion at  the  axis  of  suspension,  and  cause  the  vibra- 
tions to  be  made  in  the  vacuum  of  an  air-pump,  the 
motion  will  continue  for  many  hours. 

From  these  and  other  similar  experiments,  as  well 
as  from  numerous  analogous  observations,  we  infer 
that  if  the  interfering  forces  could  be  annihilated, 
the  motions  would  take  place  in  conformity  with 
the  law. 

II.  The  law  of  the  coexistence  or  independence  of  motions. 

The  relative  motions  of  a  system  of  bodies  are  not  af- 
fected by  any  motion  which  is  common  to  all  the  points  of 
the  system. 

To  aid  the  student  in  the  conception  of  this  law, 
suppose  a  number  of  bodies  to  be  in  motion  on  a 
surface,  plane  or  curved,  and  conceive  the  surface  to 
be  at  the  same  time  moving  parallel  to  itself;  then, 
according  to  this  law,  the  motions  of  the  bodies  on 


12  MECHANICS. 

the  surface,  or  their  relative  motions,  will  not  be 
affected  by  the  motion  which  they  have  in  common 
with  the  surface,  but  will  take  place  in  the  same 
manner  as  if  the  surface  were  at  rest.  The  surface 
having  been  introduced  merely  to  render  the  dis- 
tinction between  the  relative  motions  of  the  bodies 
and  their  common  motion  more  obvious,  it  may  now 
be  withdrawn,  if  in  its  place  we  give  to  each  of  the 
bodies  a  motion  identical  with  the  common  motion. 
We  shall  thus  have  a  system  of  bodies  moving  in 
space  in  the  manner  supposed  in  the  enunciation. 

A  case  less  general  than  the  preceding,  but  one  to 
which  it  will  be  necessary  frequently  to  refer,  is 
that  in  which  we  suppose  a  point  to  be  moving  on  a 
straight  line,  and  the  line  to  be  carried  at  the  same 
time  with  a  uniform  rectilinear  motion  parallel  to 
itself;  the  motion  of  the  point,  according  to  this 
law,  taking  place  as  it  would  were  the  line  at  rest. 

An  apt  illustration  of  this  law  is  furnished  by  a 
vessel  sailing  in  a  given  direction,  in  which  the  mo- 
tions of  bodies  relative  to  the  parts  of  the  vessel 
take  place  precisely  in  the  same  manner  as  if  the 
vessel  were  at  rest,  though,  in  addition  to  their  re- 
lative motions,  the  bodies  all  have  the  same  motion 
as  the  vessel.  Since  the  vessel  may  obviously  be 
regarded  merely  as  the  means  of  communicating 
motion  to  the  bodies,  we  may  suppose  it  to  be  with- 
drawn, if  at  the  same  time  we  suppose  the  bodies 
to  retain  their  common  motion. 


INTRODUCTION.  13 

III.   The  law  of  the  equality  of  action  and  reaction. 

To  every  action,  there  is  always  opposed  an  equal  re- 
action. 

According  to  this  law,  when  a  body  rests  upon  an 
immovable  plane,  the  pressure  of  the  body  upon  the 
plane  causes  in  the  plane  a  reaction,  in  virtue  of 
which  the  jDlane  may  be  said  to  j)i*ess  upwards  with 
the  same  force  that  the  body  presses  downwards. 
In  like  manner,  when  a  body  falls  upon  an  immov- 
able plane,  the  plane  reacts  with  a  force  equal  and 
opposite  to  that  with  which  it  is  struck  by  the  body. 

Also,  when  two  bodies,  moving  in  the  same  line, 
come  in  collision,  the  effects  upon  the  bodies,  pro- 
perly estimated,^  are  equal,  and  take  place  in  opposite 
directions. 

*  The  method  of  estimathig  the  effects  will  be  explained  in  a  subse- 
quent article. 


PART  FIRST. 


STATICS, 


8.  By  the  term  material  point,  or  'particle,  is  meant 
the  smallest  conceivable  portion  of  matter.  In  me- 
chanics, any  one  material  point  is  conceived  to  be 
exactly  similar  to  every  other  material  point.  A 
body  may  be  regarded  as  a  collection  of  material 
points. 

In  a  force,  three  things  are  to  be  considered  :  its 
point  of  application,  its  direction,  and  its  intensity. 

The  point  of  application  of  a  force  is  the  material 
point  on  Avhich  it  immediately  acts. 

The  direction  of  a  force  is  the  direction  in  which 
it  tends  to  move  its  point  of  application. 

The  intensity  of  a  force  is  its  capacity  of  producing 
motion. 

In  considering  the  effects  of  forces,  we  first  sup- 
pose the  body  upon  which  the  forces  act  to  be 
reduced  to  a  single  material  point.  This  point  we 
regard  as  entirely  passive,  and  subject  only  to  the 
action  of  the  forces  under  consideration. 

Two  forces  are  regarded  as  equal  in  intensity, 
when,  acting  separateh'  u])()U  two  material  pomts. 


STATICS.  15 

they  cause  the  points  to  describe  equal  spaces  in  equal 
times. 

Two  equal  forces  applied  to  the  same  point,  and 
acting  in  the  same  direction,  constitute  a  double 
force  ;  three  equal  forces,  a  triple  force ;  four  a 
quadruple  force,  and  so  on.  Taking  one  of  the 
equal  forces  for  the  unit  of  intensity,  and  repre- 
senting it  by  1  or  by  p,  the  intensities  of  the  double, 
triple,  quadruple,  etc.  forces,  will  be  represented  by 
2,  3,  4,  etc.,  or  2p,  Sp,  4:p,  etc.  The  intensities  of 
forces  may  thus  be  represented  by  numbers  or 
algebraic  symbols,  and  may  be  made  the  subject  of 
arithmetical  and  algebraic  operations. 

The  intensities  of  forces  may  also  be  represented 
by  straight  lines;  a  given  straight  line  being  as- 
sumed to  represent  the  unit  of  intensity,  and  a 
double,  triple,  quadruple,  etc.  force  being  repre- 
sented by  a  line  of  double,  triple,  quadruple,  etc. 
the  length  of  the  assumed  line. 

According  to  the  second  law  of  motion,  double, 
triple,  quadruple,  etc.  forces  cause  the  material 
points  to  which  they  are  a23plied  to  describe,  in  a 
given  time,  spaces  double,  triple,  quadruple,  etc.  the 
space  due  to  the  unit  of  intensity.  The  spaces 
described  by  the  material  points  are  thus  directly 
proportional  to  the  intensities  of  the  forces ;  and 
consequently  the  same  lines  which  represent  the 
intensities  of  the  forces  may  also  be  employed  to 
represent  the  spaces  described  by  the  material 
points  to  which  the  forces  are  applied. 


16  MECHANICS. 

If,  for  example,  the  lines  AB,  AC,  AD  [Fig.  1], 
represent  the  intensities  P,  Q  and  E;  of  three  forces 
referred  to  a  common  unit,  they  may  also  be  em- 
ployed to  represent  the  spaces  which  these  forces 
severally  would  cause  a  material  point  to  describe 
in  a  given  time. 

The  converse  also  is  obviously  true,  viz.  that  if 
AB,  AC  and  AD  represent  the  spaces  which  three 
forces,  whose  intensities  are  P,  Q  and  R,  would 
severally  cause  a  material  point  to  describe  in  a 
given  time,  the  same  lines  may  also  be  employed  to 
represent  the  intensities  P,  Q  and  B  respectively. 

9.  Conceive  now  a  material  point  to  be  moving 
uniformly  on  the  straight  line  AB'  [Fig  1],  and 
suppose  the  line  to  be  at  the  same  time  moving 
parallel  to  itself  with  a  uniform  rectilinear  motion  ; 
then  by  the  second  law  of  motion,  the  m^otion  of 
the  point  on  the  line  will  be  entirely  unaffected 
by  the  motion  which  it  has  in  common  with  the 
line :  and  hence  if  we  represent  the  motions  of 
the  point  and  line  during  any  time  t  by  AB  and 
AC  respectively,  at  the  end  of  that  time  the  point 
will  evidently  be  at  D,  having  described  the  diago- 
nal AD  of  the  parallelogram  constructed  on  AB 
and  AC.  But  if  we  abstract  the  line  A'B',  and 
the  motion  which  the  point  has  in  common  with 
it,  and  suppose  the  point  to  be  moving  alone  in 
space  in  the  direction  A'B',  at  the  same  rate  as 
before,  and  to  receive  when  it  arrives  at  A  an 
imj)ulse,   such   as  would   cause   it,   were  it  at  rest. 


STATICS.  17 

to  describe  the  line  AC  in  the  time  t,  the  final  result 
will  evidently  be  unaffected ;  and  in  virtue  of  the 
two  motions,  the  point,  at  the  end  of  the  time  t, 
will  still  be  found  at  D.  Again,  if,  instead  of  sup- 
posing the  point  to  be  in  motion  when  it  receives 
the  impulse  in  the  direction  AC,  we  suppose  it  to  be 
at  rest  at  A,  and  to  receive  at  the  same  instant 
another  impulse,  such,  as  acting  alone,  would  cause 
it  to  describe  in  the  time  t  the  line  AB,  it  is  still 
obvious  that  in  virtue  of  the  two  impulses  the  point 
will,  as  before,  describe  with  a  uniform  motion  the 
diagonal  of  the  parallelogram  ABDC,  and  at  the 
end  of  the  time  t  be  found  at  D. 

Let  now  the  intensities  of  the  forces  to  which  the 
motions  AB  and  AC  are  due,  be  denoted  by  P  and 
Q ;  and  as  the  motion  from  A  to  D  may  also  be 
attributed  to  a  single  force,  let  the  intensity  of  that 
force  be  denoted  by  R  :  then  the  forces  P,  Q,  and  R* 
may  be  represented  by  the  lines  AB,  AC  and  AD  ; 
and  instead  of  saying  the  forces  P,  Q  and  P,  we  may 
say  the  forces  AB,  AC  and  AD.  P,  or  AD,  is  called 
the  resultant  of  P  and  Q,  or  of  AB  and  AC ;  and  P 
and  Q,  or  AB  and  AC,  are  called  the  components  of  P, 
or  AD.  The  characteristic  property  of  the  resultant 
is  that  it  may  be  substituted  for  the  components. 
Thus  the  effect  upon  the  point  A  is  precisely  the 
same  wdiether  the  components  P  and  Q  act  upon  it 
in  the  directions  AB  and  AC,  or  the  resultant  P  acts 
upon  it  in  the  direction  AD. 

*Hereafcer,  for  the  sake  of  brevity,  we  shall  say  simply  the  force  P, 
the  force  Q,  etc.,  instead   of  the  force  whose  intensity  is  P,  Q,  etc. 

3 


18  MECHANICS. 

The  result  at  which  we  have  arrived,  may  now 
be  thus  enunciated : 

The  resultant  of  two  forces  applied  to  a  material  point, 
and  represented  by  lines  measured  from  this  point  on  their 
directions,  is  represented  in  magnitude  and  direction  hy 
the  diagonal  of  the  parallelogram  constructed  on  these 

lines. 
This  proposition  is  called  ih.Q  parallelogram  of  forces. 

The  forces  P  and  Q  may  evidently  be  supposed 
to  be  either  finite,  or  infinitely  small. 

If  A  be  a  fixed  point,  P  and  Q  will  exert  upon  it 
a  pressure  of  longer  or  shorter  duration,  the  measure 
of  which  is  evidently  Rf^ 

10.  In  figure  2,  in  which  AI  is  taken  equal  to  AB 
or  P,  as  the  angle  BxiC  diminishes,  ID  tends  con- 
stantly to  become  equal  to  BD  or  Q,  and  hence  R 
to  become  equal  to  P+  Q. 

In  figure  3,  in  which  AI  is  taken  equal  to  AD  or 
R,  as  the  angle  BAG  increases  to  180',  BI  tends 
constantly  to  become  equal  to  BD  or  Q,  and  R  to 
become  equal  to  P  —  Q. 

Hence,  when  the  forces  P  and  Q  act  in  the  same 
straight  line,  it  may  be  inferred  from  the  above 
proposition  that  the  resultant  is  equal  to  their  sum 
or  difierence,  according  as  they  act  in  the  same  or 
opposite  directions.  This  result  may  also  be  deduced 
immediately  from  the  second  law  of  motion. 

*When  P  and  Q  are  forces  of  the  kind  commonly  called  impulsive, 
such,  for  example,  as  that  exerted  in  driving  a  nail  with  a  hammer,  the 
duration  of  the  pressure  is  very  brief,  though  finite  :  when  they  are  of 
the  same  nature  as  gravity,  the  pressure  is  continuous. 


STATICS.  19 

When  the  forces  act  in  opposite  directions,  we  have 
12  =  P  — Q. 

But  P  may  be  considered  as  the  resultant  of  two 
forces  P'  and  p,  both  acting  in  the  same  direction  as 
P  ;  and  p  as  the  resultant  of  two  others  P"  and  p, 
both  acting  in  the  same  direction  as  p,  and  so  on. 
The  same  is  also  true  of  Q ;  hence  we  have 

R  z=  P'-\-P"  +  P"'+kc.—  Q'—Q"—Q"'—&c [1] 

When  the  forces  P',  P" ,  etc.,  Q',  Q",  etc.,  are  in 
equilibrium,  we  have  11=0  ;  and  hence 

p/  -j.  pn  _|_  pi"  -j_  &c.  —    Q'+  Q'  +  Q'"  +  &c. 

is  the  equation  which  expresses  the  condition  of 
equilibrium  of  any  number  of  forces  which  act  in 
the  same  straight  line. 

11.  Resuming  the  consideration  of  the  parallelo- 
gram of  forces  :  Since  [Fig.  4]  BD  is  equal  to  AC,  we 
have  the  three  forces  P,  Q,  and  R  represented  by  the 
sides  of  the  triangle  ABD.  Moreover  we  perceive 
that  the  angle  at  B  is  the  supplement  of  the  angle 
BAG  formed  by  the  directions  of  the  components  P 
and  Q,  and  that  the  angle  ADB  is  equal  to  the  angle 
DAG  which  the  direction  of  Q  makes  with  that  of 
R.  Hence  of  the  three  forces  P,  Q  and  R,  and  the 
angles  comprehended  between  their  directions,  any 
three  being  given,  one  of  the  three  at  least  being 
a  force,  the  remaining  three  can  be  determined  by 
solving  the  triangle  ABD. 


20  MECHANICS. 

Thus  denoting  the  angles  which  the  directions  of 
P  and  Q  make  with  the  direction  of  R,  by  b  and  c, 
and  the  angle  which  they  make  with  each  other  by 
a  ;  if  P,  Q  and  a  are  given,  and  R  and  b  are  required, 
to  determine  the  latter,  we  have 

R'  =  P  +Q—2PQ    cosB 
—  p^  _|_  Q'^  _|_  2  PQ  •  cos  a, 
and  R     :     Q     ::     sina     :     sin  b. 

12.  When  three  forces  P,  Q  and  R'  (Fig.  4),  are 
in  equilibrium  about  a  point  A,  any  one  of  them,  as 
R\  must  evidently  be  equal  and  directly  opposite  to 
the  resultant  of  the  other  two.  But  employing  the 
same  notation  as  in  the  preceding  articles,  and  denot- 
ing the  supplements  of  b  and  c  hy  b'  and  c',  we  have 


and 


and  hence,  since  R  =  R', 

R'    :    P    :     Q    : :     sin «    :     sin  c'    :     sin  b'. 

That  is,  when  three  forces  are  in  equilibrium  about  the 
same  point,  each  of  them  may  be  represented  by  the  sine 
of  the  angle  comprehended  between  the  directions  of  the 
other  two, 

13.  Any  number  of  given  forces  P,  P',  P ',  P"\ 
etc.,  applied  at  a  point,  and  situated  in  the  same  or 
in  different  planes,  may,  by  means  of  the  parallelo- 
gram of  forces,  be  reduced  to  a  single  force.  We  first 
determine  the  resultant  R  of  any  two  of  them,  as  P 


R 

: 

P 

sma 

sin  c 

sin  a 

sin  c', 

R 

: 

Q 

sin  a 

sin  6 

sin  a 

sin  b' ; 

STATICS.  21 

and  P' ;  then  the  resultant  R\  of  R  and  any  one  of 
the  remaining  forces,  as  P"  ;  and  so  on,  till  all  the 
given  forces  have  been  compounded  :  the  last  re- 
sultant will  be  the  resultant  of  the  system. 

Figure  5  presents  a  case  in  which  four  forces  P,  P', 
P",  P",  are  reduced  to  a  single  force  R",  An  inspec- 
tion of  the  figure  shows  that  the  resultant  may  be 
determined  by  the  following  construction  :  The  lines 
AB,  AB',  AB'/,  AB'",  representing  the  four  forces  : 
through  the  point  B,  draw  BC  equal  and  parallel  to 
AB';  through  C,  draw  CD  equal  and  parallel  to  AB"; 
and  through  D,  draw  DE  equal  and  parallel  to  AB": 
the  straight  line  joining  the  points  E  and  A  is  the 
resultant  required.  This  method  of  finding  the  re- 
sultant is  applicable  to  any  number  of  forces. 

We  have  just  seen  that  any  number  of  forces 
applied  at  a  point  may  be  reduced  to  a  single  force, 
or  resultant.  The  converse  also  is  evidently  true,  viz. 
that  a  single  force  applied  at  a  point  may  be  resolved 
into  any  number  of  forces,  or  components,  all  acting 
upon  the  same  point,  and  producing  the  same  effect 
as  the  single  force. 


14.  Methods  of  determining  the  position  of  a  point 

A  convenient  and  elegant  method  of  treating  tht- 
subject  of  forces,  consists  in  decomposing  the  forces 
into  separate  systems,  parallel  to  certain  assumed 


22  MECHANICS. 

straight  lines.  The  application  of  this  method  re- 
quires a  knowledge  of  the  first  principles  of  analytic 
geometry,  of  which  we  shall  here  give  a  brief 
exposition. 

§1.  Method  of  determining  the  position  of  a  point 
on  a  line. 

The  position  of  a  point  on  a  given  straight  line 
XX'  [Fig.  6],  is  evidently  determined  when  we  know^ 
its  distance  from  an  assumed  point  0  of  the  line,  and 
the  direction  in  which  the  distance  is  to  be  laid  off 
from  the  assumed  point.  The  method  of  indicating 
the  direction  is  conventional,  and  consists  in  affecting 
the  distance  with  the  sign  plus  or  minus,  according 
as  the  point  is  situated  to  the  right  or  left  of  0. 
Thus,  employing  the  letter  x,  as  the  general  symbol 
denoting  the  distance,  the  equation 

X  =z  -\-  a 

determines  a  point  M,  situated  at  the  right  of  0,  at 
a  distance  from  it  equal  to  a  linear  units ;  and  the 
equation 

a;  =  —  a 

determines  a  point  M',  situated  at  the  same  distance 
from  0,  in  the  opposite  direction. 

§  2.  Method  of  determining  the  position  of  a  point 
on  a  plane. 

The  position  of  a  point  on  a  plane  is  determined 
when  we  know  its  distances,  affected  with  their 
proper  signs,  from  two  straight  lines  drawn  in  the 
plane  at  right  angles  to  each  other. 


STATICS.  23 

Let  XX',  YY'  [Fig.  7]  represent  these  lines  form- 
ing by  their  intersection  the  four  right  angles  XOY. 
YOX',  X'OY',  Y'OX.  Consider  the  point  M  situated 
in  the  angle  XOY,  and  draw  through  it  the  straight 
lines  MP,  MQ,  parallel  respectively  to  the  lines  YY  , 
XX',  and  intersecting  them  in  the  points  P  and  Q. 
Then  it  is  evident  that  when  P  and  Q  are  given,  the 
point  M  can  be  determined  by  drawing  through 
them  the  straight  lines  MP,  MQ,  parallel  to  YY , 
XX',  respectively ;  it  being  at  the  intersection  of 
these  lines.  But  P  and  Q  are  given  when  the  dis- 
tances OP  and  OQ,  or  MQ  and  MP,  are  given  :  thus 
the  point  M  is  determined  by  its  distances  from  the 
lines  Xr,  YY'. 

The  line  OP  or  MQ  is  called  the  abscissa,  and  OQ 
or  MP  the  ordinate,  of  the  point  M.  The  abscissa  and 
ordinate,  taken  together,  are  called  the  co-ordinates 
of  M.  The  line  XX'  is  called  the  axis  of  abscissas, 
and  YY'  the  axis  of  ordinates,  and  together  they  are 
called  the  axes  of  co-ordinates.  The  point  0  in  which 
the  axes  intersect,  is  called  the  origin  of  co-ordinates. 
The  general  symbols,  commonly  employed  to  denote 
the  abscissas  and  ordinates,  are  x  and  y ;  and  hence 
the  axis  XX'  is  frequently  called  the  axis  of  x,  and 
YY'  the  axis  of  y.  The  axis  XX'  is  usually  supposed 
to  be  horizontal. 

The  quantities  x  and  y  are  supposed  to  be  capable 
of  assuming  all  possible  values,  both  positive  and 
negative ;  and  according  to  the  method  of  the  pre- 
ceding section  [§  1],  the  positive  values  of  x  are  laid 


24  MECHANICS. 

off  on  XX'  from  0  to  the  right,  the  negative  vaUies 
from  0  to  the  left;  the  positive  vahies  of  y  are  hiicl 
off  on  YY  from  0  upwards,  the  negative  values  from 
0  downwards.  Thus,  denoting  the  values  of  x  and  y, 
m  any  particular  case,  by  a  and  h, 

whon  X  —  -\-  a  and  rj  —  -\-  h,  the  point  is  in  the  1st  angle  ; 

"    X  =  —  a  and  y  =  +  6,  "  "  2d      "      ; 

"    X  r=  —  a  and   ?/  :=  —  ^,  "  "  3d      "       ; 

«'    X  z=z  -\-  a  and  ?/  =  —  ^,  "  "  4th    " 

Wheux=:  -|-  a  and  ?/  =        0,  the  point  is  on  the  axis  of  x  ; 

«    X  =        0  and  y  =z  -{-  b,  "  "  "  y, 

"     X  =        0  and  y  =3        0,  the  point  is  at  the  origin. 

§  3.  Method  of  determining  the  position  of  a  point 
in  space. 

The  position  of  a  point  in  space  is  determined 
when  we  know  its  distances,  affected  with  their 
proper  signs,  from  three  planes  draw^n  through  an 
assumed  point  at  right  angles  to  each  other. 

Let  XOY,  XOZ,  YOZ  [Fig.  8]  be  the  three  fixed 
planes  intersecting  each  other,  when  sufficiently  pro- 
duced, in  the  lines  XX',  YY',  ZZ',  and  determining 
by  their  intersection  eight  trihedral  angles.  Consider 
the  point  M  situated  in  the  angle  0-XYZ,  and 
draw  through  it  three  planes  parallel  to  the  fixed 
planes,  viz  : 

MM\iM'"  parallel  to  the  plane  XOZ,  and  cutting  the  line  YY'  in  Q; 
AJ  M'PM"  parallel  to  the  plane  YOZ,  and  cutting  the  line  XX' in  P; 

and 
MM"RM'''  parallel  to  the  plane  XOY,  and  cutting  the  line  ZZ'  in  R. 

Then  it  is  evident  that  the  point  M  can  be  deter- 
mined, when  the  points  Q,  P  and  R  are  known ;  for, 


STATICS.  25 

drawing  through  these  points  three  planes  parallel 
respectively  to  the  fixed  planes,  they  will  intersect 
each  other  at  the  point  M.  But  the  points  P,  Q  and 
R  are  known  when  the  distances  OP,  OQ  and  OR, 
or  the  equal  lines  MM",  MM",  MM',  are  given  :  thus 
the  point  M  is  determined  in  position  by  its  dis- 
tances from  the  three  fixed  planes. 

The  distances  OP,  OQ  and  OR  are  called  the  co- 
ordinates of  the  point  M;  the  lines  XX',  YY',  ZZ', 
are  called  the  axes  of  co-ordinates;  and  the  planes 
XOY,  XOZ,  YOZ,  are  called  the  co-ordinate  planes.  The 
point  0  is  called  the  origin.  The  co-ordinates  OP,  OQ 
and  OR  are  denoted  by  the  general  symbols  x,  y  and 
z  respectively ;  and  hence  the  fixed  planes  are  deno- 
minated the  planes  of  xy,  xz  and  yz  respectively. 
The  plane  of  xy  is  generally  supposed  to  be  horizon- 
tal. The  positive  values  of  x,  y  and  z  are  laid  off, 
according  to  the  conventional  method,  on  the  several 
axes  from  0  towards  the  points  X,  Y  and  Z  respec- 
tively, and  the  negative  values  from  0  in  the  opposite 
directions. 

The  signs  of  the  co-ordinates  of  a  point  determine 
in  which  of  the  eight  angles  the  point  is  situated. 
Thus,  denoting  the  numerical  values  of  x,  y  and  z, 
in  a  particular  case,  by  a,  h  and  c : 

When  X  zz:  «,    i/  =z  b,  and  2  =  c,  the  point  is  situated  in  the 

angle  0  -  XYZ  ; 
when  X  =  — a,   y  z=z  — 6,  and  z  z=z  — c,  it  is  situated  in  the 

angle  0  -  X'Y'Z'. 

When  one  of  the  co-ordinates  is  zero,  the  point  is 

4 


26  MECHANICS. 

situated  in  the  plane  of  the  other  two  co-ordinates ; 
thus  when  x  —  a,y  —  h,  and  z  =  0,  the  point  is  in  the 
plane  of  xy.  When  two  of  the  co-ordinates  are  each 
equal  to  zero,  the  point  is  on  the  axis  of  the  third 
co-ordinate ;  thus  when  x  =  a,  y  =  0,  and  2-  =  0,  it 
is  on  the  axis  of  x.  When  x  =  0,  y  —  0,  and  z  =  0, 
the  point  is  at  the  origin. 


Of  forces  situated  in  the  same  plane,  and  applied  at 
the  same  point. 

15.  Let  0  [Fig.  9]  be  the  point  of  application  of 
the  forces ;  and  through  it,  in  the  plane  of  the  forces, 
let  the  co-ordinate  axes  XX',  YY',  be  drawn. 

The  direction  of  a  force  P  is  determined  by  the 
angles  which  it  makes  with  OX  and  OY,  the  parts 
of  the  axes  on  which  the  positive  co-ordinates  are 
laid  off,  the  angles  being  reckoned  on  both  sides  of 
these  lines  from  0  to  180^.  Thus,  considering  the 
four  distinct  cases  which  can  occur,  as  represented 
in  figures  9,  10,  11,  12  : 

"When  the  direction  of  the  force  is  situated  in  the  angle  XOY,  its 
position  is  determined  by  the  angles  XOM,  YOM  ; 

when  in  the  angle  YOX',  it  is  determined  by  the  angles 
XOM',  YOM'; 

when  in  the  angle  X'OY',  it  is  determined  by  the  angles 
XOM",   YOM"; 

when  in  the  angle  XOY',  it  is  determined  by  the  angles 
XOM"',   YOM". 


STATICS.  27 

Let  now  the  force  P,  in  the  several  cases,  be  repre- 
sented by  the  equal  lines  OM,  OM',  OM",  OM"  ;  and 
let  it  be  decomposed  into  the  components  OB  or 
OB',  OC  or  OC,  coincident  respectively  with  the  two 
axes.  Then  we  perceive  that  the  components  coin- 
cident with  XX'  act,  when  the  force  is  situated  in 
the  1st  or  4th  angle,  from  0  to  the  right ;  when  in 
the  2d  or  3d  angle,  from  0  to  the  left :  also  that  the 
components  coincident  with  YY'  act,  when  the  force 
is  situated  in  the  1st  or  2d  angle,  from  0  upwards ; 
when  in  the  3d  or  4th,  from  0  downwards. 

When,  therefore,  two  or  more  forces,  acting  upon 
the  point  0,  are  decomposed  in  directions  coincident 
with  the  axes,  there  may  be  two  sets  of  components 
in  the  direction  of  each  axis,  the  components  of  one 
set  acting  in  one  direction,  those  of  the  other  in  the 
direction  opposite.  Considering  the  components 
which  act  in  the  line  XX' :  if  we  affect  those  acting 
in  opposite  directions  with  opposite  signs,  as,  for 
example,  those  which  act  from  0  to  the  right,  with 
the  sign  plus  +,  and  those  which  act  from  0  to  the 
left  with  the  sign  minus  — ;  then  the  resultant  will 
evidently  be  equal  to  the  algebraic  sum  of  the  com- 
ponents. The  same  is  true  of  the  components  which 
act  in  the  line  YY'. 

But  the  values  of  the  components  determined  by 
the  usual  method,  will  necessarily  be  affected  with 
the  proper  signs.  Thus,  denoting  generally  the 
angles  which  the  direction  of  P  makes  with  OX  and 
OY  by  a  and  /3,  when  the  force  is  situated  in  the 
first  angle,  we  have 


28  MECHANICS. 

OB  =  P  cos  BOM  =  +  P  cos  a, 
OC  =  Pcos  COM-  +  Pcosi3; 

when  it  is  situated  in  the  second  angle,  we  have 

OB'  =  P  cos  B'OM'  =  —  P  cos  a, 
OC  -=  P  cos  COM'    =  +  P  cos  /3  ; 

Avhen  in  the  third,  we  have 

OB'  =  P  cos  B'OM"  =  —  P  cos  a, 

OC  =  P  cos  COM"  =  —  P  cos  /3  ; 

when  in  the  fourth, 

OB  =:PcosBOM'"  =  +  Pcosrt', 

OC  =  P  cos  COM"'  =  —  P  cos  i3.* 

Now  let  there  be  any  number  of  forces,  P,  P',  P", 
etc.  [Fig.  13],  acting  upon  the  point  0,  in  the  direc- 
tions OM,  OM',  OM",  etc.;  and  let  the  angles  which 
their  directions  make  with  the  axes  be  denoted  by 
a,  a',  a",  etc.,  /3,  ,/3',  /3",  etc. :  then  if  we  denote  the  alge- 
braic sum  of  the  components  which  act  in  XX'  by  X, 
and  that  of  the  components  which  act  in  YY'  by  F, 
we  shall  have 

Z  =  Pcos«'+P'cos«''  +  P"eos^"  +  etc., [3] 

r  =  Pcos/3+P'cos/3'4-P"cos/3"-|-etc [1] 

These  equations  are  general.  In  a  given  case,  the 
values  of  the  angles  will  determine  the  algebraic 
signs  of  the  terms  of  the  second  members.  A  com- 
mon method  of  denoting  the  algebraic  sum  of  a 
series  of  terms,  all  of  which  are  of  the  same  form,  is 
to  write  only  a  single  term  preceded  by  the  Greek 

*  In  these  expressions  of  the  value  of  the  components,  it  will  be 
observed  that  by  cos  a  and  cos  /3,  we  denote  the  numerical  values  of  the 
cosines  a  and  /3, — hence  the  algebraic  signs  are  expressed. 


STATICS.  29 

letter  i.  Applying  this  method  to  the  above  equa- 
tions, we  have 

X  =   2  (PcosrY), 
Y  —  :l  (Pcos/3). 

We  have  now  reduced  the  forces  P,  P\  P",  etc.  to 
the  two  forces  X  and  F,  acting  at  right  angles  to 
each  other ;  and  if  we  represent  X  by  OE  [Fig.  13], 
and  Y  by  OF,  the  resultant  R  of  these  two  forces  will 
be  represented  by  the  diagonal  OG  of  the  rectangle 
OEGF.  This  resultant  will  be  the  resultant  of  the 
system  P,  P',  P",  etc.,  and  will  be  determined  in 
intensity  by  the  equation 

R  —  \^X^Y~ [5] 

To  find  the  angles  a  and  b  which  the  direction  of 
R  makes  with  the  axes,  we  have 

cos  a  =z  — ,       cos  6  =1  — - [G] 

The  quantities  R,  a  and  b,  are  thus  comjDletely  de- 
termined. 

IG.   Conditions  of  equilibrium. 

In  order  that  P,  P',  P",  etc.  may  be  in  equilibrium 
about  the  point  0,  their  resultant  must  evidently 
be  equal  to  zero ;  a  condition  which  gives 

X'  +  r^  —  0. 
But  as  the  square  of  a  quantity  is  essentially  posi- 
tive, this  equation  can  be  satisfied  only  by  making 
X  and  Y  separately  equal  to  zero ;  thus  the  condi- 
tions of  equilibrium  are  expressed  by  the  equations 
A  =  0,       Y  —  0. 


30  MECHANICS. 


Of  forces  applied  at  the  same  point,  and  situated 

IN  different  planes. 

17.  Let  the  three  forces  X,  F,  Z,  be  applied  at  the 
point  0  [  Fig.  14],  and  be  represented  in  intensity 
and  direction  by  the  lines  OS,  OS',  OS".  On  these 
lines  as  adjacent  edges,  construct  the  parallelopiped 
RS.  Then  it  is  evident  that  ON  the  diagonal  of  the 
base  OSNS'  is  the  resultant  of  OS  and  OS'  that  is,  of 
X  and  F;  and  that  ON'  the  diagonal  of  the  parallelo- 
piped is  the  resultant  of  ON  and  OS",  that  is,  of  X, 
Y  and  Z.  Thus  the  resultant  of  the  forces  X,  Y  and 
Z,  represented  by  the  lines  OS,  OS',  OS',  is  the 
diagonal  of  the  parallelopiped  constructed  on  these 
lines.  If  the  directions  of  the  three  forces  intersect 
each  other  at  right  angles,  the  parallelopiped  will  be 
rectangular ;  and  denoting  the  resultant  ON'  by  R, 
we  shall  have 

R  =  \^X'+  Y'+Z\ 

The  direction  of  a  force  P  situated  in  space,  and 
applied  at  the  point  0  [Fig.  15],  is  determined  by 
the  angles  which  it  makes  with  the  lines  OX,  OY, 
OZ,  the  parts  of  the  co-ordinate  axes  drawn  through 
0,  on  which  the  positive  co-ordinates  are  laid  off,  the 
angles  being  reckoned  from  0  to  180°.  Thus  if  P  is 
situated  in  the  angle  0  -  XYZ,  its  direction  OM  is 
determined  by  the  angles  XOM,  YOM,  ZOM,  each 
angle  being  less  than  90° ;  if  it  is  situated  in  the 
angle  0  -  XYZ',  its  direction  OM^  is  determined  by 


STATICS.  31 

the  angles  XOM^,  YOM^,  ZOIVI ,  the  first  two  being 
each  less  than  90°,  the  third  greater  than  90\^ 

If  the  intensity  of  P  be  represented  by  OM,  its 
three  rectangular  components  in  the  directions  of 
the  axes  will  evidently  be  OP,  OQ  and  OH ;  and  if 
the  angles  which  it  makes  with  the  axes  be  denoted 
by  a,  13  and  y  respectively,  we  shall  have 
OP  =  OM.  cos  MOP  =  Pcosa, 
OQ  =  OM-cosMOQ  =  Pcos[3, 
OR  =  OM-cosMOR  =  Pcosy. 
Now  let  the  forces  P,  P',  P\  etc.  be  applied  at 
the  point  0,  and  let  the  angles  which  their  directions 
make  with  the  co-ordinate  axes  be  denoted  by  a,  [3, 
y;  a',  [3',  y' ;  a",  (3",  y",  etc. :  then  the  components  co- 
incident with  XX'  will  be 

Pcosa,     P'cosa',     P"cos{x",  etc.; 
those  coincident  with  YY'  will  be 

Pcos/3,     P' cos  13',     P"  cos  13",  etc.; 
and  those  coincident  with  ZZ'  will  be 

Pcosy,     P'cosy',     P"cosy",  etc.; 

and  if  the  algebraic  sums  of  the  components  in  the 
several  directions  be  denoted  by  X,  Y  and  Z  re- 
spectively, we  shall  have 

*The  angles  which  the  direction  of  P  makes  with  the  axes  OX,  OY, 
fix  its  position  on  two  conic  surfaces,  of  which  O  is  the  common  vertex, 
and  OX  and  OY  the  axes  :  these  surfaces  intersect  each  other  in  two 
straight  lines,  and  the  third  angle  determines  which  of  the  two  lines 
is  the  direction  sought.  Thus  a  certain  relation  exists  between  the 
three  angles.  In  accordance  with  our  plan,  this,  like  many  other  parts 
of  the  general  subject,  is  left  to  be  -developed  either  by  the  pupil  or  his 
instructor. 


32  MECHANICS. 

X  =  Pt'osrt'+P'cosrt''  +  P"cos«"  +  ete.=  I  (P  cos^), 

Y  =  P  cos  ,3 -\- P' cos  i3'  -\- F"  Go^  ,3"  +  etc.=  1  (P  cos /3),  i-[7] 

Z  —  P  cos  y  +  P  cos  y  +  P'  cos  y"  +  etc.=  :E  (P  cos  y),  J 

If  we  denote  bv  R  the  resultant  of  Z,  7,  Z,  that 
is,  of  P,  P',  P  ',  etc.,  we  shall  have 

R  —  VX  +  ¥'+  Z  ; 
and  if  we  denote  the  angles  which  the  direction  of 
R  makes  with  the  three  axes  by  a,  b  and  c  respec- 
tively, we  shall  have 

cos  a  z=  -^— ,       cos  o  =  ^r-,       cos  c  =  -=-  ;    [81 

K  K  K 

and  thus  the  resultant  will    be  completely  deter- 
mined.^ 

18.  Conditions  of  equilibrium. 

In  order  that  the  forces  may  be  in  equilibrium,  R 
must  be  equal  to  zero ;  a  condition  which  gives 

X'-\-Y-'+Z"-  =  0, 
and  hence 

X  =  0,     y  r3  0,    z  1=  0. 

If  the  point  0  is  not  entirely  free,  but  subject  to 
the  condition  of  remaining  on  a  given  surface,  it  is 
not  essential  to  an  equilibrium  that  the  resultant 
should  be  equal  to  zero  :  it  is  only  necessary  that 
it  should  act  towards  the  surface,  and  at  right  angles 
to  it.  The  resistance  of  the  surface  is  equal  and 
directly  opposite  to  R. 

*The  understanding  of  this  article  may  be  greatly  facilitated  by  a 
model  of  the  co-ordinate  planes,  which  the  student  himself  can  easily 
construct  of  pieces  of  pasteboard. 


STATICS.  33 

19.  As  yet  we  have  constantly  supposed  the  forces 
under  consideration  to  be  applied  at  the  same  point. 
We  shall  now  suppose  them  applied  at  different 
jDoints ;  the  points  being  conceived  to  be  so  con- 
nected, that  their  positions  with  respect  to  each 
other  are  invariable.  The  lines  of  direction  of  the 
forces  may  be  parallel  to  each  other,  or  oblique. 
We  shall  first  consider  the  case  in  which  they  a^e 
parallel. 


OF    PARALLEL    FORCES. 

20.  Let  P  and  Q  be  any  two  parallel  forces,  applied 
at  the  extremities  A  and  B  [Fig.  16]  of  an  inflexible 
straight  line  AB,  and  acting  in  the  same  sense"^  in 
the  lines  AV  and  BS,  and  let  their  intensities  be 
represented  by  the  lines  AU  and  BT. 

Without  affecting  the  action  of  these  forces,  we 
may  apply  at  A  and  B  in  AB  produced,  two  equal 
and  opposite  forces  AM  and  BN.  The  resultants 
AD  and  BI  of  the  four  forces  AM,  AU,  BN,  BT, 
may  then  be  transferred  along  their  lines  of  direction 
(produced  backwards)  to  C  their  point  of  meeting, 
and  there  be  resolved  into  the  four  forces  CG,  CL, 

*When  the  lines  of  direction  of  forces  are  parallel,  we  use  the  word 
serine  to  indicate  the  relations  of  the  directions  :  thus  when  two  parallel 
forces  act  towards  the  same  parts  of  space,  both  to  the  right  or  both  to 
the  left,  for  example,  we  say  they  act  in  the  same  sense;  when  they 
act  towards  opposite  parts  of  space,  we  say  they  act  in  oiJj)Osite  senses. 
When  we  speak  of  the  directions  of  parallel  forces,  we  mean  their  lines 
of  direction. 

5 


34  MECILVNICS. 

CH,  CK,  equal  and  parallel  respectively  to  the  four 
forces  applied  at  A  and  B;  but  CH  and  CG  being 
equal,  and  acting  in  the  same  line  and  in  opposite 
directions,  destroy  each  other  :  hence  the  resultant 
of  the  system  is  CL  +  CK,  or  P  +  Q,  acting  in  the 
same  sense  as  the  given  forces,  and  in  the  line  CO 
parallel  to  their  lines  of  direction. 

The  resultant  may  evidently  be  supposed  to  be 
applied  at  0,  where  its  direction  intersects  the  line 
AB.  To  determine  this  point,  we  have  from  the 
similar  triangles  AUD,  COA,  BTI,  COB, 

AU    :    UD    ::    CO    :    OA, 
BT     :    TI     ::    CO    :    OB; 

or  UD  X  CO  =  AU  X  OA 

TI  X  CO  =  BT  X  OB; 
'and  hence  P  X  OA  =   Q  X  OB, 

or  P    :    Q    ::    OB    :    OA. 

That  is,  0,  the  point  of  application  of  the  resultant, 
divides  the  line  AB  into  parts  reciprocally  proportional 
to  the  intensities  of  the  component  forces. 

The  above  proportion,  it  will  be  perceived,  is  in- 
dependent of  the  angle  which  the  directions  of  the 
forces  make  with  AB,  and  is  true  for  any  straight 
line  drawn  through  0  and  terminated  by  their 
directions. 

Resuming  the  proportion, 

P    :    Q    ::    OB    :    OA, 
we  get  by  composition, 

P    :    P+  Q    ::    OB    :    OB  +  0A; 


STATICS.  35 

or,  denoting  P  +  Q  hy  R, 

P    :    R    ::    OB    :    AB. 

In  like  manner  we  get 

Q    :    R    ::    OA    :    AB; 
and  hence  we  have 

P    :    Q    :    R    ::    OB    :    OA    \    AB. 

If  the  force  P  be  represented  by  OB,  the  forces  Q 
and  R  will  be  represented  bj  OA  and  AB  respec- 
tively. 

Thus  each  of  the  three  forces  may  he  represented  by  the 
fart  0/  AB  intercepted  by  the  directions  of  the  other  two. 

The  decomposition  of  a  single  force  into  two  pa- 
rallel forces,  which  shall  act  in  the  same  sense  and 
satisfy  certain  conditions,  is  a  very  simple  applica- 
tion of  the  foregoing  results.  Thus,  suppose  it  be 
required  to  resolve  the  given  force  R  into  the  two  pa- 
rallel forces  P  and  Q,  one  of  which  {P)  is  also  given. 
Here,  assuming  the  points  A,  B  and  0  [Fig.  17],  as 
the  points  of  application  of  the  forces  P,  Q  and  R 
respectively,  the  given  quantities  are  R,  P  and  OA ; 
and  the  required  quantities,  Q  and  OB.  To  find  Q, 
we  have R  =  P  +  Q;  from  which,  Q  =  R  —  P ;  and 
to  find  OB  we  have 

Q    :    P    ::    OA    :    OB; 
from  which 

OB  =  ^  X  OA  =  ^^  X  OA. 

21.  We  shall  now  consider  the  case  in  which  the 
forces  P  and  Q,  applied  as  before  at  the  extremities 
of  the  line  AB  [Fig.  18],  act  in  opposite  senses. 


36  MECHANICS. 

The  force  Q  may  be  either  greater  or  less  than  P 
if  Q  is  greater  than  P,  let  it  be  resolved  into  thi 
two  forces  P'  and  R,  both  acting  in  the  same  sense 
as  Q;  the  first  equal  to  P,  and  applied  at  A  in  the 
direction  of  P  produced.  We  have,  to  determine 
the  intensity  of  R, 

Q  =  P+  R; 
from  which, 

R  =    Q  —  P    —    Q  —  P  ; 

and  to  determine  0  its  point  of  application,  we  have 

R    :    P'  —  P    ::    Kl^    :    OB; 
from  which, 

OB  =  Q^  X  AB. 

Employing  now  in  place  of  Q,  its  two  components 
P'  —  P  and  R,  the  two  given  forces  P  and  Q  may 

be  replaced  by  the  three  P,  P'  and  R  ;  of  these,  P 
and  P  destroying  each  other,  there  remains  only 
the  force  R  as  the  equivalent  of  the  given  forces  : 
hence  R  —  (Q  —  P)  applied  at  the  point  0,  and  act- 
ing in  the  same  sense  as  Q,  is  the  resultant  sought. 
In  like  manner,  it  may  be  shown  that  if  P  is 
greater  than  Q,  the  resultant  R'  is  equal  to  P  —  Q, 
acting  in  the  same  sense  as  P,  and  applied  at  0'  a 
point  situated  to  the  left  of  A. 

Thus  when  the  given  forces  act  in  opposite  senses, 
their  resultant  is  situated  without  the  directions  of  the 
forces  (not  between  them,  as  in  the  preceding  case), 
and  on  the  side  of  the  greater  force,  and  it  acts  in  the  same 
sense  as  the  greater. 


STATICS.  37 

In  this,  as  in  the  preceding  case,  it  is  evidently 
true  that 

P    :    Q    :    R    ::    OB    :    OA    :    AB. 

22.  If  in  the  value  of  OB  [Fig.  18],  derived  from 
the  proportion 

P    :    R    ::    OB    :    AB, 

we  substitute  for  R  its  value  (Q  —  P),  we  get 

OB  =  Q^  X  AB, 

From  this  equation,  it  appears  that  as  the  difference 
between  Q  and  P  diminishes,  the  resultant,  con- 
stantly dimimishing,  is  applied  at  a  point  more  and 
more  distant  from  B,  till  when  Q  ==  P  its  intensity 
is  reduced  to  zero,  and  the  distance  of  the  point  of 
application  becomes  infinite.  In  this  case,  there- 
fore, there  is  really  no  resultant.  Such  a  sj^stem 
consisting  of  two  equal  and  parallel  forces  acting  in 
opposite  senses,  but  not  in  the  same  straight  line,  is 
called  a  couple. 

23.  Of  any  number  of  'parallel  forces  applied  at  points 
having  any  position  whatever. 

Let  P,  P'  P",  etc.  denote  the  forces,  and  let  their 
points  of  application  be  A,  B,  C,  etc.  [Fig.  19];  the 
points  being  so  connected  by  the  inflexible  straight 
lines  AB,  BC,  etc.  as  constantly  to  retain  the  same 
relative  positions.  Required  the  intensity  of  the  re- 
sultant, and  its  point  of  application. 

We  first  compound  any  two  of  the  forces,  as  P 
and  P',  and  find  the  point  M  at  which  their  resultant 
P  +  P'  is  applied,  by  the  proportion 


38  MECHANICS. 

AB    :    AM    : :    P  +  P    :    P . 

We  next  compound  the  resultant  P  +  P  with  any 
one  of  the  remaining  forces,  as  P  ;  and  find  the 
point  N  at  which  their  resultant  P  +  P'  -\-  P"  is 
applied,  by  the  proportion 

MC    :    MN    : :    P  +  P'  -{-  P'    :    P". 

Proceeding  in  this  manner,  constantly  compounding 
the  last  resultant  with  one  of  the  remaining  forces, 
we  finally  arrive  at  K  the  point  of  application  of  the 
resultant  R  z=^  P  -\-  P'  +P"  +  etc.  of  the  whole  sys- 
tem. The  resultant  evidently  acts  in  the  same  sense 
as  the  components,  and  parallel  to  their  common 
line  of  direction. 

24.  In  the  case  just  considered,  we  have  supposed 
all  the  forces  to  act  in  the  same  sense.  When  some 
of  them  act  in  one  sense,  and  some  in  the  opposite 
sense,  a  circumstance  indicated  by  difference  of  sign,  we 
find  by  the  preceding  method  the  resultants  i?  and 
R'  of  the  two  systems  and  their  points  of  applica- 
tion K'  and  K"  separately ;  and  then  reduce  these 
resultants  to  a  single  force  R,  and  find  its  point  of 
application  K  by  the  ordinary  rules.  When  the  two 
resultants  are  equal,  but  not  directly  opposite,  the 
case  is  that  of  Art.  22. 

25.  In  determining  the  point  of  a^Dplication  of  the 
resultant  of  the  system,  we  employ  only  the  intensi- 
ties of  the  forces,  and  the  mutual  distances  of  their 
points  of  application ;  quantities  which  are  not  af- 
fected by  giving  to  the  directions  of  the  forces  a 


STATICS.  39 

common  motion  about  these  points.  Hence,  what- 
ever change  we  make  in  the  common  direction  of  a 
system  of  parallel  forces,  applied  at  points  of  w  hich 
the  relative  positions  are  invariable,  by  revolving 
the  directions  of  the  forces  about  their  points  of 
application  by  a  common  motion,  the  point  of  appli- 
cation of  the  resultant  of  the  system  remains  the 
same.  The  point  of  a23plication  of  the  resultant  of 
a  system  of  parallel  forces  is  called  the  centre  of 
'parallel  forces. 

If  the  centre  of  parallel  forces  in  any  sj^stem  is  a 
fixed  point,  the  system  is  evidently  in  equilibrium 
in  all  positions  about  that  point. 

26.  Let  us  now  suppose  the  points  of  application 
of  the  forces  to  be  referred  to  three  co-ordinate 
planes,  and  let  us  seek  the  position  of  the  point  of 
application  of  the  resultant,  or  the  centre  of  parallel 
forces,  with  respect  to  these  planes. 

Let  X,     y,     z,  be  the  co-ordinates  of  the  point  A 

B 
C 


x\   y,  z. 

(( 

ii. 

X",  y",  z". 

(( 

a 

etc. ;  ' 

and  X,,    ?/ ,    z,   the  co-ordinates  of  the   centre   #f 
parallel  forces. 

Let  E  [Fig.  20]  be  the  point  of  application  of  the 
resultant  of  P  and  P';  draw  AI,  EK  and  BL  perpen- 
dicular to  the  plane  XOY ;  join  the  points  I  and  L, 
and  draw  BG  parallel  to  IL.  We  have  [Art.  20] 
P  +  P'    :    P    ::    AB    :    EB. 


40  MECHANICS. 

But  the  similar  triangles  AGB,  EHB  give 
AB    :    EB    ::    AG    :    Ell: 

hence  P  +  P'     :       i^     : :     AG     :     EH; 

and  (P  +  P')  EII  =  P  X  AG. 

Adding  to  each  member  of  this  equation  the  product 

(P  +  F)  HK,  we  get 

(P  +  P')  (HK +EII)  =   (P+POHIv  +  P  X  AG 

==  P  (HK  +  AG)  +P'X  UK, 
or  (P  +  P')EK  =  PXAI  +  P'XBL; 

and  denoting  the  ordinate  EK  by  Z,  we  have 

(P  -\-P')Z  =  Pz  +  P'z'. 

Compounding  the  resultant  P  -{- P  applied  at  E, 
with  the  third  force  P\  and  denoting  the  co- 
ordinate of  the  point  of  application  of  their  resultant 
by  Z',  we  get 

{P+P'+  P")  Z'  —   (P  +  P')  Z  +  P"z"  ; 

and  substituting  for  {P  -\- P')  Z  its  value  Pz  +  P'z\ 
w^e  have 

(P_|_p/_|_p//)Z'  =  Pz  +  P'z'  -^P"z". 

Proceeding  in  the  same  manner  with  the  remain- 
ino;  forces,  and  denotino;  as  before  the  resultant  of 

C3  '  c5 

tjie  system  by  R,  we  finally  arrive  at  the  equation 

Rz,  z=i  Pz  +  P'z  +  P"z"  +  P"'z'"  +  etc [9] 

The  first  member  of  this  equation  is  the  product 
of  the  resultant  by  the  perpendicular  let  fall  from 
its  point  of  application,  upon  the  plane  XOY  ;  and 
the  second  member  is  the  sum  of  the  products  ob- 
tained by  multiplying  each  of  the  given  forces  by 


STATICS.  41 

the  perpendicular  let  fall  from  its  point  of  applica- 
tion, upon  the  same  plane.  These  products  are 
called  the  moments  of  the  forces  with  respect  to  the 
plane  XO  Y ;  thus  Rz,  Pz,  are  called  the  moments  of  R 
and  P  respectively  referred  to  this  plane.  The 
result  contained  in  this  equation  may  then  be  thus 
enunciated  :  The  mo?nent  of  the  resultant  of  a  system  of 
parallel  forces,  with  respect  to  any  plane  whatever,  is  equal 
to  the  sum  of  the  moments  of  the  forces  with  respect  to  the 
same  plane. 

Since  the  forces  are  affected  with  the  positive  or 
negative  sign,  according  to  the  sense  in  which  they 
act,  and  the  signs  of  the  co-ordinates  of  the  points  of 
application  are  also  positive  or  negative,  according 
to  the  positions  of  the  points,  the  signs  of  the 
moments,  determined  by  the  ordinary  rules,  may  be 
either  positive  or  negative. 

If  the  moments  of  the  given  forces  be  taken  with 
respect  to  the  planes  XOZ,  YOZ,  we  shall  get 

Ry,  =  Py  +  P'y'  +  P"y"  -f  etc., [10] 

Rx,  1=  Pz  +  PV  -f  P"x"  +  etc. ; [11] 

and  thus  we  shall  have,  to  determine  the  centre  of 
parallel  forces, 

__  Px  -\-  P'xf  +  P"x"  +  etc. 
h  -  ^  , 

_  Py  +  P'y^  +  P'^y'f  +  etc. 
^'  ~  R 

Pz  +  P'zf  +  P"z"  +  etc . 
z,  — 


R 

6 


42  MECHANICS. 

27.  If  the  moments  he  taken  with  respect  to  a  plane 
which  passes  through  the  centre  of  parallel  forces,  the  sum 
of  the  moments  will,  in  that  case,  he  equal  to  zero ;  for  if 
we  suppose  the  plane  XOY  to  be  that  plane,  we 

shall  have 

z,  =z  0, 

and  hence  Pz  +  P'z'  +  F"z"  +  etc.  —  0. 

28.  The  points  of  application  in  the  same  plane. 

It  may  happen  that  the  points  of  application  of 
the  forces  are  all  in  the  same  plane ;  when  this  is 
the  case,  the  centre  of  parallel  forces,  if  it  exist,  is 
also  situated  in  this  plane,  and  two  co-ordinates  are 
sufficient  to  determine  its  position.  Thus  if  we 
suppose  the  plane  XOY  to  coincide  with  this  plane, 
we  shall  have 

2  =  0,     z'  =  0,     z"  =  0,  etc., 
and  hence  z,  z=z  0  ; 

and  the  position  of  the  point  will  be  determined  by 
the  co-ordinates  x^  and  y^,  the  values  of  which  are 
given  by  equations  [10]  and  [11]. 

28'.   The  points  of  application  in  the  same  straight  line. 

If  the  points  of  application  are  all  situated  in  the 
same  straight  line,  the  centre  of  parallel  forces  will 
also  be  found  in  this  line,  and  its  position  will  be 
determined  by  a  single  co-ordinate.  Thus  if  we 
suppose  the  axis  OX  to  coincide  with  this  line,  we 
shall  have 


STATICS.  43 

y  =  0,     r/'  =:  0,     y  =  0,  etc.  ; 
2=0,     z'   —  0,     z"   =  0,  etc.  ; 
and  hence  y^  =  0  and  z,  r=  0, 

and  the  value  of  x^,  given  by  the  equation 

Rx,  =  Px  +  PV  +  P"x"  +  etc., 
will  determine  the  position  of  the  centre  of  parallel 
forces  with  respect  to  the  point  0. 

29.   Conditions  of  equilibrium. 

Let  the  plane  XOY  [Fig.  21]  be  taken  perpendi- 
cular to  the  direction  of  the  forces ;  and  let  the  forces 
be  reduced,  as  in  Art.  24,  to  the  partial  components 
R'  and  R\  applied  at  the  points  K'  and  K ".  The 
conditions  of  equilibrium  will  obviously  be, 

§  1.  That  R'  and  R"  shall  be  equal  to  each  other, 
and  act  in  opposite  senses. 

§  2.  That  their  lines  of  direction  shall  coincide. 
The  first  condition  is  expressed  by  the  equation 

R'  z=z  —R" [a] 

But  supposing  P,  P',  P",  etc.  to  be  the  components 
of  R',  and  P",  P'\  P\  etc.  to  be  those  of  R'\  we  have 

R'    z=  P    +P'  +P"  +  etc., 
and  R"  —  P'"  +  P^^  +  P^+etc. : 

hence,  by  substitution,  equation  [a]  becomes 

P  +  P'+P"+P'"  +  Pi^'  +  P^+etc.  =  0 [12] 

To  express  the  second  condition  algebraically,  let 
the  co-ordinates  of  K'  and  K",  referred  to  the  planes 
XOZ,  YOZ,  be  denoted  by  x.^,  y^^  and  x  ^^,  y^^^  :  then 
in  order  that  the  lines  of  direction  of  R'  and  R"  may 
coincide,  we  must  evidently  have. 

X,,  =  7,,,  and  y„  —  7j,„, 


44  MECHANICS. 

Multiplying  these  equations  by  equation  [a],  we  get 

R'x„  =  —R"x,„, [b] 

R'lj,   =  -R'y,„ [c] 

But  by  the  principle  of  moments  [Art.  26],  we  have 

R'x„     —  Px       +P'x'     +P"x"+etc., 
R"x,„   —  P"'x"'  +  Pi^'x'^-  +  P'-x^-f  etc.  ; 
R'y„     =  Py       +P'y'     +P"3/'  +  etc., 
R"y,„  —  P'"y"'-\-P'Hf'  +  P'if+QiQ, 

Hence  equations  [b]  and  [c]  become  by  substitution, 

Px  -f  P'x'  +  P"x"  +  P'"x"'  +  P-x'^-  +  P'x'  +  etc.  =  0,     [13] 
Py-\-P'y'-\-P"7j"  +  P'"y"'  +  P'''y''-{-P'if-\-eiQ.   —  0     [14] 

Thus  when  the  system  is  in  equilibrium,  equations 
[12],  [13]  and  [14]  must  be  satisfied;  and  conversely 
when  these  equations  are  satisfied,  the  system  is  in 
equilibrium.  These  equations  express, 

§  1  [Equa.  12].  That  the  algebraic  sum  of  the  forces 
7nust  be  equal  to  zero. 

§.  2  [Equa.  13  and  14].  That  the  sum  of  the  moments 
of  the  forces,  taken  with  respect  to  two  planes  at  right 
angles  to  each  other,  and  parallel  to  the  common  direction 
of  the  forces,  must  also  be  equal  to  zero. 

30.  When  the  second  of  these  conditions  only  is 
satisfied,  that  is,  when  equations  [13]  and  [14]  are 
satisfied,  but  not  equation  [12],  the  given  forces  will 
have  a  resultant,  the  direction  of  which  Avill  coincide 
with  the  intersection  of  the  two  planes  with  respect 
to  which  the  moments  are  taken ;  for  since  the  sum 
of  the  moments  of  the  forces  is  equal  to  zero  with 
respect  to  eacli  of  tlio  ])lnnes.  the  moments  of  the 


STATICS.  45 

resultant  with  respect  to  these  planes  are  also  equal 
to  zero,  and  hence  the  centre  of  parallel  forces  must 
be  in  each  plane ;  moreover  the  direction  of  the 
resultant  is  parallel  to  the  planes  :  consequently  the 
direction  of  the  resultant  must  coincide  with  the 
intersection  of  the  planes. 

If  there  be  a  fixed  point  in  the  intersection  of  the 
planes,  the  resultant  will  be  destroyed,  and  an  equi- 
librium will  exist.  Hence  when  parallel  forces  which 
have  a  resultant  are  applied  to  a  system  of  material  points 
rigidly  connected^  of  lohich  one  is  fixed,  there  will  he  an 
equilibrium,  if  the  su?n  of  the  moments  of  the  forces  taken 
with  respect  to  two  planes  drawn  through  the  fixed  point, 
parallel  to  the  direction  of  the  forces,  and  at  right  angles 
to  each  other,  is  equal  to  zero. 

We  shall  next  consider  the  subject  of  oblique  forces 
applied  at  different  points.  There  are  two  cases  : 
V  When  the  forces  act  in  the  same  plane  ;  2" 
When  they  act  in  different  planes. 


Of  oblique  forces  applied  at  different  points,  and 

ACTING  in  the  SAME  PLANE. 

31.  Let  the  intensities  of  the  forces  be  denoted  by 
P,  F ,  P" ,  P",  etc. ;  and  let  the  plane  which  contains 
their  directions  be  taken  for  the  co-ordinate  plane 
XOY  [Fig.  22].  Let  the  points  A,  B,  C,  D,  etc.,  con- 
nected with  each  other  in  an  invariable  manner,  be 
the  points  of  application  of  the  forces  as  represented 


46  MECHANICS. 

in  the  figure.  The  resultant  of  the  system,  when  it 
exists,  may  be  found  in  the  following  manner  :  Com- 
mencing with  P  and  P',  any  two  of  the  forces,  pro- 
duce their  directions  till  they  meet  in  G.  Suppose 
the  two  forces  applied  at  this  point ;  and  taking  Qcm 
and  Qn  to  represent  their  intensities,  construct  the 
parallelogram  GmQ;'n  :  the  diagonal  GG'  is  their 
resultant.  In  like  manner,  compound  the  resultant 
GG'  with  P"  any  one  of  the  remaining  forces, 
transferring  their  points  of  application  to  H  their 
point  of  meeting,  and  thus  determine  their  resul- 
tant HH'.  Continue  this  process  till  all  the  forces 
have  been  compounded  :  the  last  resultant  will  be 
the  resultant  of  the  system.  In  the  case  of  the 
given  forces  P,  P',  P",  P",  the  resultant  is  II'. 

If,  in  any  part  of  the  operation,  we  arrive  at  forces 
of  which  the  directions  are  parallel,  we  apply  to  them 
the  rules  for  parallel  forces. 

If,  in  the  final  result,  we  find  two  equal  forces  act- 
ing in  parallel  lines  and  opposite  senses,  the  system 
has  no  resultant  [Art.  22] . 

The  above  method  is  equivalent  to  applying  the 
given  forces  at  the  point  I,  in  lines  parallel  to  their 
original  directions,  and  then  reducing  them  to  a 
single  force.  For  the  resultant  II',  applied  at  I,  may 
be  resolved  into  the  two  forces  In"  =  P"  and  Ini' ; 
Im"  may  be  resolved  into  lu  and  Iv,  equal  and  paral- 
lel to  the  forces  H;z'  =  P"  and  llm' ;  and  so  on  till 
in  place  of  II'  we  have  all  the  original  forces  applied 


STATICS.  47 

at  I,  and  acting  in  lines  parallel  to  their  primitive 
directions. 

32.  The  resultant  of  the  forces  P,  P',  P\  P",  etc. 
may  also  be  determined  by  the  following  method : 
The  co-ordinates  of  the  points  of  ajDplication  referred 
to  the  axes  OX,  OY  being  given,  and  also  the  angles 
which  the  directions  of  the  forces  make  with  the 
axes,  let  each  force  be  resolved  into  two  components 
parallel  to  OX  and  OY  respectively.  The  whole 
system  of  forces  will  thus  be  resolved  into  two  sys- 
tems of  components,  the  one  parallel  to  OX,  the 
other  parallel  to  OY.  Let  these  components  be 
applied  at  the  points  where  their  directions  intersect 
the  axes,  and  let  the  resultants  X  and  Y  of  the  two 
23artial  systems  be  determined  in  intensity  and  posi- 
tion by  the  rules  for  parallel  forces  [articles  24  and 
28]  :  the  resultant  of  X  and  F,  or  the  resultant  of 
the  whole  system,  may  then  be  determined  by  the 
ordinary  rule. 

Before  we  proceed  to  determine  the  conditions  of 
equilibrium,  it  is  necessary  to  make  known  some  of 
the  pro23erties  of  what  are  called  moments  j^ef erred  to 
a  point. 

33.  Of  moments  referred  to  a  point. 

Let  P  and  P'  be  any  two  forces  meeting  in  A 
[Fig.  23],  and  represented  by  AB  and  AC  respec- 
tively, and  let  their  resultant  AD  be  denoted  by  R. 
In  the  plane  of  the  forces,  assume  any  point  0 ;  and 


48  MECHANICS. 

from  it  let  fall  upon  the  directions  of  P,  P',  and  R, 
the  perpendiculars  01,  01',  01 '.  Through  the  points 
A  and  0,  draw  the  straight  line  AO,  and  through  A 
draw  AQ  at  right  angles  to  AO.  Also  from  the 
points  B,  C  and  D,  let  fall  upon  AQ  the  perpendicu- 
lars BD',  CD",  DD  ";  and  through  C  draw  CE  paral- 
lel to  AQ.  The  similar  triangles  AOI,  ABD'  give 

AO    :    01    ::    AB    :    AD'; 
or  denoting  AO  and  01  by  c  and  p  respectively, 
c    :   p    ::    P    :    AD'; 

from  which  Ave  get 

AD'  ^  It. 
c 

In  like  manner,  denoting  by  p'  and  r  the  perpen- 
diculars 01'  and  01",  and  employing  the  correspond- 
ing triangles,  we  get 

AJ>"  =  ^^,    AD"'  =  ^. 
c  c 

But  the  equal  triangles  ABD',  CDE,  give 

AD'    =  CE    =D"D'", 
and  hence  we  have 

AD'"  =  AD"  +  D"D"' 
z=  AD"  +  AD'; 
and  substituting  in  this  equation  the  values  of  the 
several  terms  found  above,  we  get 

Rr^  _    Pp  P'p' 

c  c     '^      c     ^ 

or  Rr  z:^   Pp-\-  P'p' [15] 

We  have  supposed  the  point  0  to  be  taken  with- 
out the  angle  BAG,  or  the  opposite  vertical  angle 


STATICS.  49 

B'AC  :  when  it  is  taken  within  either  of  these  angles, 
BAG  for  example,  as  in  figure  24,  we  have,  as  in  the 
preceding  case, 

AD'     =  CE  =  D"D'", 
and  hence  Ti'D"'  =  AD", 

Hence  we  have 

AD'"    z=  AD'— D'D"'  zn  AD'  — AD"; 

and  substituting  as  above,  we  get 

Rr       =  P/j  —  P'p' [16] 

The  products  Rr,  Pp,  etc.  are  called  the  moments  of 
the  forces  R,  P,  etc.  with  respect  to  the  assumed  point 
0  ;  and  the  point  itself  is  called  the  centre  of  moments. 
Thus  the  moment  of  a  force  ivith  respect  to  a  point,  is  the 
product  of  the  force  by  the  perpendicular  let  fall  from  the 
point  upon  the  direction  of  the  force.  Employing  these 
terms,  the  principle  involved  in  equations  [15]  and 
[16]  may  be  thus  enunciated  : 

The  moment  of  the  resultant  of  any  two  forces  is  equal 
to  the  sum  or  difference  of  the  moments  of  the  forces,  accord- 
ing as  the  centre  of  moments  is  taken  without  or  within  the 
angle  made  by  the  directions  of  the  forces,  or  the  opposite 
angle  at  the  vertex  made  by  their  directions  produced. 

Another  enunciation  of  this  principle  may  be  given 
by  introducing  the  idea  of  motion.  Thus,  suppose 
the  perpendiculars  01,  01',  01",  to  be  inflexible  lines, 
capable  of  motion  in  the  plane  of  the  forces,  about 
the  point  0  regarded  as  a  fixed  point ;  then  the 
forces  P,  P'  and  R,  which  we  may  imagine  applied 
at  the  points  I,  I'  and  I",  will  tend  to  give  to  the 


50  MECHANICS. 

perpendiculars  a  motion  of  rotation  about  0 ;  and 
we  perceive  that  in  figure  23,  in  which  the  centre  of 
moments  is  without  the  angle  BAG,  or  its  opposite 
anaie  at  the  vertex,  these  forces  will  tend  to  turn 
their  points  of  application  in  the  same  sense  about 
0  :  while,  on  the  contrary,  in  figure  24,  in  which  the 
centre  of  moments  is  within  one  of  these  angles,  the 
two  components  will  tend  to  turn  their  points  of 
application  in  opposite  senses  about  0,  and  the  resul- 
tant will  tend  to  turn  its  point  of  application  in  the 
same  sense  as  the  component  which  has  the  greater 
moment.  We  may  therefore  say  that  the  moment  of 
the  resultant  of  any  two  forces  is  equal  to  the  sum  or  dif- 
ference of  the  moments  of  the  forces,  according  as  the 
forces  tend  to  turn  their  points  of  application  in  the  same 
sense,  or  in  opposite  senses,  about  the  fixed  point  assumed 
in  their  plane  as  the  centre  of  moments. 

34.  This  principle  can  be  shown  to  be  true,  what- 
ever the  number  of  forces.  Let  us  consider  the  case 
in  which  the  centre  of  moments  is  so  situated  that 
the  first  three  of  the  forces  P,  P',  P ",  P  ",  etc.  tend 
to  turn  the  system  in  one  sense,  and  the  remaining 
forces  in  the  opposite  sense  about  that  point.  Let 
Q  be  the  resultant  of  P  and  P',  and  Q'  that  of  Q  and 
P  '.  Also  let  p,  p  ,  p" ,  q  and  q'  denote  the  perpendi- 
culars let  fall  from  the  centre  of  moments  on  the 
dh^ections  of  P,  P',  P",  Q  and  Q'  respectively.  Then, 
according  to  the  principle  just  demonstrated,  we 
shall  have 


STATICS.  51 

Qq     =   Pi?  +  P'p', 
Q'q'   r=    Qq  +  Py  ; 

and  substituting  in  the  second  of  these  equations  the 
value  of  Qq  derived  from  the  first,  we  shall  get 

Q'q'  —  Pp  +  P'p'  +  P"j)". 
Again,  denoting  by  Q^  the  resultant  of  the  remaining 
forces  P"\  P'\  etc. ;  by  q^  the  perpendicular  let  fixll 
from  the  centre  of  moments  upon  its  direction ;  and 
by  p"\  p'^,  etc.  the  perpendiculars  drawn  from  the 
same  point  to  the  directions  of  P'",  P'^,  etc.,  we  shall 
have 

Q,q,    =    P"y  +  Plv^iv_|_g(.c 

Let  now  the  resultant  of  Q'  and  Q^  (that  is,  the  re- 
sultant of  the  given  forces  P,  P\  P",  etc.)  be  denoted 
by  R,  and  the  perpendicular  let  fall  from  the  centre 
of  moments  upon  its  direction  by  r.  Moreover  let 
it  be  recollected  that  Q'  and  Q^  tend  to  turn  their 
points  of  application  in  opposite  directions.  Then, 
according  as  Q'q  is  greater  or  less  than  Q^q^,  we  shall 
have,  since  Rr  must  be  positive, 

Rr  =    Q'q'—Q^q^^ 
or  Rr  =   Q,q,  —  Q'q'. 

In  the  first  case,  the  force  R  will  tend  to  turn  the 
system  in  the  same  sense  as  the  force  Q'  and  conse- 
quently in  the  same  sense  as  the  forces  P,  P'  and  P", 
Supposing  this  to  be  the  case  in  question,  and  substi- 
tuting for  Q'^',  Qq^ ,  the  values  found  above,  we  get 

Rr  —  Pp  +  P'p'  4-  P"p"  —  P"'p"'  —  pivpiv  _  etc- [17] 


52  MECHANICS. 

Thus,  whatever  the  number  of  forces,  the  moment 
of  the  resultant  is  equal  to  the  sum  of  the  moments  of  the 
forces  which  tend  to  turn  the  system  in  the  same  sense  as 
the  resultant,  minus  the  sum  of  the  moments  of  the  forces 
which  tend  to  turn  it  in  the  opposite  sense. 

35.  If,  in  the  phane  of  a  system  of  forces  such  as 
we  have  been  considering,  there  is  a  fixed  point  with 
which  the  points  of  application  are  connected,  and 
about  which  they  may  revolve  in  the  plane  of  the 
system,  the  forces  will  obviously  be  in  equilibrium  if 
their  resultant  passes  through  this  point.  Suppose 
their  resultant  thus  to  pass,  and  take  the  fixed  point 
for  the  centre  of  moments ;  then  we  shall  have 

r      =  0, 
and  hence  Rr  z=  0, 

and  equation  [17]  will  become 

Pp  +  P'p'  -\-P"p"  +  etQ-  z=i  0. 
Hence,  in  order  to  an  equilibrium  in  this  case,  it  is 
only  necessary  that  the  sum  of  the  moments  of  the  forces 
which  tend  to  turn  the  system  in  one  sense  about  the  fixed 
point,  should  be  equal  to  the  sum  of  the  moments  of  the 
forces  which  tend  to  turn  the  system  in  the  opposite  sense, 
the  moments  being  taken  ivith  respect  to  the  fixed  point. 

36.  If,  in  the  case  of  a  system  of  forces  P,  P' ,  P", 
etc.  in  which  there  is  a  resultant,  that  is,  in  which  R 
is  not  equal  to  zero,  we  have 

Pp  +  Pp  +  P"p"  +  etc.  =  0, 
then  R?'  nz  0, 

and  hence  r      =0, 


STATICS.  53 

and  consequeiitl}^  tlie  resultant  passes  through  the 
point  assumed  as  the  centre  of  moments. 

37.  Conditions  of  equilihrium. 

Let  P,  P\  and  P"  he  any  three  forces  situated  in 
the  same  23lane,  and  applied  at  the  points  A,  B  and 
C,  as  represented  in  figure  25.  It  is  essential  to  an 
equilibrium,  that  their  directions  should  meet  in  the 
same  point ;  for  in  order  that  any  two  of  them,  as 
P  and  P',  may  be  in  equilibrium  with  the  third  P", 
the  latter  must  act  in  the  same  line  with  the  resul- 
tant R  of  the  two  former,  and  hence  its  direction 
must  pass  through  the  point  A  at  which  the  direc- 
tions of  P  and  P'  intersect.  But  P  may  be  supj)osed 
to  be  the  resultant  of  two  other  forces  P  '  and  P"', 
applied  at  the  point  E,  as  in  the  figure ;  P'"",  of  two 
others  P^  and  P^\  and  so  on.  Hence  in  order  that 
any  number  of  forces  situated  in  the  same  plane,  and 
applied  at  different  points,  may  be  in  equilibrium, 
they  must  be  reducible  to  three  forces  which  meet  in 
the  same  point.  To  express  this  condition  algebrai- 
cally, let  the  forces  P,  P  and  P '  be  supposed  to  be 
applied  at  A  [Fig.  26]  their  point  of  meeting,  and  let 
their  intensities  be  represented  by  AB,  AC  and  AD  , 
AD^  bping  equal  to  AD  or  R  the  resultant  of  P  and 
P  .  Also  from  a  point  0  assumed  in  the  plane  of 
the  forces, 'let  the  lines  01,  01',  01",  be  drawn  per- 
pendicular to  the  directions  of  P,  P'  and  P  or  R. 
Recurring  to  article  33,  and  employing  the  same 
notation  as  in  that  article,  we  perceive  that  the  rela- 
tion between  these  forces  and  perpendiculars  is 
expressed  by  the  equation 


54  MECHANICS. 

Rr  —   Pp±P'p' , 

or,  substituting  for  Rr  its  equal  P"'p'\  by  the  equa- 
tion 

P"p"   —   Pp  ±  P'p' ; [aj 

the  upper  or  lower  sign  of  the  second  member  being 
employed,  according  to  the  position  of  the  point  0. 
This  equation  expresses  that  the  moment  of  the 
force  which  tends  to  turn  the  system  in  one  sense 
about  the  centre  of  moments,  is  equal  to  the  sum  of 
the  moments  of  the  forces  which  tend  to  turn  it  in 
the  opposite  sense ;  and  a  little  consideration  renders 
it  apparent  that  when  this  condition  is  fulfilled,  the 
forces  P,  P'  and  P"  must  meet  in  the  same  point. 
Equation  [a]  may  therefore  be  taken  for  the  equa- 
tion of  condition  in  the  case  of  three  forces,  the  forces 
necessarily  meeting  in  the  same  point  when  it  is 
satisfied.  To  deduce  from  it  the  equation  for  a 
greater  number  of  forces,  let  P  be  considered  as  the 
resultant  of  the  two  forces  P  "  and  P''',  applied  at 
some  point  E  taken  in  the  direction  of  P  :  we  shall 
have,  by  the  principle  of  moments, 
Pp  —  p'"^'"_piyv^ 
(it  being  supposed,  as  represented  in  the  figure,  that 
P  '  and  P'''  tend  to  turn  their  points  of  application 
in  opposite  senses  about  the  point  0,  P  and  P  "  in 
the  same  sense ; )  and  substituting  this  value  of  Pp 
in  equation  [a],  and  employing  the  upper  sign  of  the 
second  member,  we  shall  get 

or  P"p"  +  jP  y  V  =  Py    +  P"'p"', 


STATICS.  55 

for  the  equation  of  condition  in  the  case  of  four 
forces.  By  thus  successively  regarding  one  of  the 
forces  as  the  resultant  of  two  others,  the  equation  of 
condition  may  be  found  for  any  number  of  forces- 
The  general  equation  is  commonly  written  thus, 

Pp  -J-  p^y  _|_  p'pn  _|_  p  upn,  ^  gtc.    1=    0  ; 

the  signs  of  the  terms  being  determined  by  the  direc- 
tions in  which  the  several  forces  tend  to  turn  their 
i3oints  of  application. 

Since  this  equation  indicates  that  the  forces  P, 
P',  P",  P"\  etc.  are  reducible  to  three  forces  which 
meet  in  the  same  point,  when  it  is  satisfied,  we  may 
suppose  all  the  forces  applied  at  the  point  of  meeting 
of  the  three  forces  in  lines  parallel  to  their  primitive 
directions.  The  conditions  of  equilibrium  Avill  thus 
be  reduced  to  those  of  article  15;  and  adopting  the 
notation  of  that  article,  we  shall  have  for  the  remain- 
ing equations  of  condition, 

P  cosa-j^P'  cosa'-\-P"cosa"  -\-P"'  Qosa"'-\-etc.  =  0, 
Pcos/3+ P' cos/3'4-P"cos/J"4-P'"  cos/3'"-|-etc.  =  0. 

Thus  the  equations  of  condition  for  the  equilibrium 
of  any  number  of  forces  situated  in  the  same  plane, 
and  applied  at  different  points,  are 

1  {P  cosa)  =  0, [-18J 

2(Pcos/3)  =  0,    [19] 

^{PP)  =  0 [20] 


56  MECHANICS. 

Of  forces  applied  at  different  points,  and  situated 
IN  different  planes. 

38.  This  is  the  most  general  case  :  We  shall  not 
discuss  it  in  detail,  but  merely  indicate  the  principal 
steps  in  the  several  processes. 

The  intensities  and  directions  of  the  forces  being 
given,  and  also  the  positions  of  their  points  of  appli- 
cation (these  points  being  conceived  to  be  rigidly 
connected  as  in  the  preceding  case),  to  determine 
their  resultant,  when  it  exists,  we  proceed  as  follows : 

1"  We  reduce  the  given  system  to  two  partial 
systems  (b)  and  (c)  :  the  one  consisting  of  forces  si- 
tuated in  the  j^lane  of  xy  ',^  the  other,  of  forces  per- 
pendicular to  that  plane,  and  hence  parallel  to  the 
axis  of  z. 

2*^  By  the  foregoing  methods  [Articles  32  and  23], 
we  determine  the  resultants  of  the  systems  (b)  and  (c), 

3°  When  these  resultants  are  situated  in  the  same 
plane,  we  reduce  them  to  a  single  force  by  the  or- 
dinary rule. 

If  the  resultants  are  not  situated  in  the  same 
plane,  the  given  forces  are  obviously  not  reducible 
to  a  single  force. 

39.  Conditions  of  equilibrium. 

If  the  systems  (b)  and  (c)  are  separately  in  equili- 
brium, the  given  system  must  evidently  be  in  equili- 
brium. The  converse  is  also  true,  viz.  that  if  the 
given  system  is  in  equilibrium,  the  systems  (b)  and 

*The  position  of  the  co-ordinate  planes  being  arbitrary,  we  may- 
give  to   the  plane  .ry  any   position   whatever. 


STATICS.  57 

(c)  must  be  separately  in  equilibrium.  For  if  an 
equilibrium  exists  among  the  given  forces,  it  will 
evidently  not  be  destroyed  by  supposing  any  line  of 
the  plane  of  xy,  connected  with  the  points  of  appli- 
cation, to  become  immovable;  but  in  this  case,  the 
forces  situated  in  the  plane  of  xy  will  be  destroyed 
by  the  resistance  of  the  fixed  line ;  and  hence  the 
forces  parallel  to  the  axis  of  z^  unless  in  equilibrium 
amongst  themselves,  will  tend  to  turn  the  plane 
about  this  line  :  therefore,  since  the  equilibrium 
must  continue,  the  forces  parallel  to  the  axis  of  z 
must  destroy  each  other.  The  system  [c]  being  thus 
necessarily  in  equilibrium  of  itself,  the  system  [b] 
must  be  also.  The  conditions  of  equilibrium  for 
these  systems  have  already  been  found  [articles  29 
and  37] ;  those  for  the  system  [b]  are  expressed  by 
the  equations  [18],  [19]  and  [20] ;  those  for  the  sys- 
tem [c],  by  the  equations  [12],  [13]  and  [14]. 

40.  When  the  system  contains  a  fixed  line,  about 
which  the  points  of  application  may  revolve,  without 
being  capable  of  moving  in  a  direction  parallel  to  it 
(in  the  same  manner  as  the  material  points  of  a  solid 
body  about  an  axis  on  which  the  body  is  prevented 
from  sliding),  the  conditions  of  equilibrium  can  be 
expressed  by  a  single  equation.  This  case  of  equili- 
brium being  one  to  which  we  shall  have  to  refer 
hereafter,  we  will  treat  it  more  fully  than  we  have 
treated  the  preceding  case. 

Let  OZ  [Fig.  27]  be  the  fixed  line  or  axis,  about 
which  the  points  of  application  may  revolve  in  the 


58  MECHANICS. 

manner  just  supposed,  and  let  it  be  taken  for  the 
axis  of  z.  Let  AK  be  the  direction  of  any  one  of  the 
forces,  as  P,  and  A  its  point  of  application.  Through 
AK  let  a  plane  be  drawn  perpendicular  to  the  plane 
of  xy ;  and  let  P,  represented  by  AK,  be  decomposed 
in  this  plane  into  the  two  forces  AH  and  AG,  the 
one  parallel  to  the  plane  of  xy,  the  other  parallel  to 
the  axis  of  z.  Since  the  point  A  can  by  hypothesis 
have  no  motion  parallel  to  the  axis  of  z,  the  second 
component  must  be  destroyed.  The  first,  which  we 
will  denote  by  Q,  is  expended  in  tending  to  give  to 
the  point  Ji.  a  motion  of  rotation  about  the  axis. 
We  will  now  show  that  this  force  may  be  replaced 
by  another,  equal  and  parallel  to  it,  but  applied  in 
the  plane  of  xi/.  For  this  purpose,  let  the  line  GA 
be  produced  to  meet  the  plane  of  xi/  in  A' ;  and 
through  A',  let  the  line  AH'  be  drawn  parallel  to 
AH.  Also  through  the  axis  OZ,  let  the  plane  ON'NN" 
be  drawn  perpendicular  to  the  parallels  AH,  AH', 
and  meeting  these  parallels  produced  in  N  and  N  . 
Without  affecting  the  system,  we  may  apply  at  the 
point  A'  two  forces  S  and  S',  each  equal  to  Q,  and 
acting  in  the  opposite  directions  AH,  A'E ;  thus  re- 
placing the  force  Q  by  the  forces  S^  S'  and  Q.  But 
conceiving  the  plane  ON'NN  "  to  be  rigidly  connected 
with  the  points  of  aj^plication,  the  forces  Q  and  S\ 
which  we  may  suppose  applied  at  N  and  N',  and 
which  tend  to  turn  this  plane  in  opposite  directions, 
will,  as  we  shall  presently  show,  destroy  each  other 
by  means  of  the  fixed  axis ;  and  we  shall  have,  as 


STATICS.  59 

the  equivalent  of  Q,  only  the  force  S,  equal  and 
parallel  to  Q,  and  acting  in  the  same  sense  as  it.  Each 
force  of  the  system  is  capable  of  a  similar  reduction. 
Thus  the  given  forces  P,  P',  P",  etc.,  applied  at  the 
points  A,  B,  C,  etc.,  may  be  replaced  by  the  coin])o- 
nents  Q,  Q',  Q",  etc.,  Q ,  Q  ,  Q^,  etc.,  applied  at 
determinate  points  of  the  plane  xij.  Hence  since 
the  point  0  is  fixed,  in  order  to  an  equilibrium,  it  is 
only  necessary  [Art.  35]  that  the  sum  of  the  moments 
of  the  forces  which  tend  to  turn  the  system  in  one 
sense,  may  be  equal  to  the  sum  of  the  moments  of  the 
forces  which  tend  to  turn  it  in  the  opposite  sense 
about  the  point  0  taken  as  the  centre  of  moments. 
If  then  we  denote  the  perpendiculars  drawn  from 
0  to  the  directions  of  the  forces,  by  q,  q',  q",  etc., 
?/'  ^//'  ^//y'  ^^^-  respectively,  and  suppose  that  the 
forces  Q,  Q',  Q ",  etc.  tend  to  turn  the  system  in  one 
sense,  and  Q ,  Q  ,  Q  ,  etc.  in  the  opposite  sense,  we 
shall  have  for  the  equation  of  condition, 

Qq  -|_  Q'q'  _|_  etc.  —  Q,q,  —  Q„q„  —  etc.    =:    6 [21] 

To  show  that  the  forces  Q  and  8'  are  in  equilibrium 
with  each  other,  by  means  of  the  fixed  axis,  at  the 
points  0  and  N'',  let  the  forces  S"  and  S"\  S'''  and 
S""  be  applied,  each  equal  and  parallel  to  Q;  S"  and 
S'""  acting  in  the  same  sense  as  Q,  S'"  and  S""  in  the 
opposite  sense.  These  forces  will  obviously  not 
change  the  state  of  the  system  ;  and  hence  we  may 
consider  the  two  forces  Q  and  S'  as  replaced  by  the 
six  forces  Q  and  S\  S'  and  S\  S'"  and  .S^'^  But  of 
these,  Qand   ,S^",  ,S^'  and  S""  mav  be  reduced  to  two 


60  MECHANICS. 

equal  and  opposite  forces  applied  at  I  the  point  of 
intersection  of  the  diagonals  ON,  N'N",  and  are  there- 
fore in  equilibrium ;  and  the  remaining  forces  S'" 
and  S''\  acting  upon  the  fixed  points  0  and  N",  are 
destroyed  by  the  reaction  of  these  points.  Conse- 
quently the  six  forces  are  in  equilibrium  by  means 
of  the  fixed  axis,  and  hence  the  equivalent  system 
Q  and  S'  must  also  be  in  equilibrium. 


CENTRE    OF    GRAVITY. 

41.  The  force  which  causes  a  body,  when  not  sup- 
ported, to  fall  to  the  earth,  is  called  gravity.  We 
learn  from  experiment,  I''  That  gravity  acts  with 
equal  intensity  upon  the  particles  of  all  bodies,  how- 
ever the  bodies  may  difier  in  size,  form,  or  nature  ; 
2°  That  it  acts  in  directions  perpendicular  to  the 
surface  of  the  earth,  or  the  surface  of  a  liquid  at  rest. 

The  line  of  direction  of  gravity  at  any  place,  is 
called  the  vertical  at  that  place  ;  and  any  plane  per- 
pendicular to  the  vertical,  is  called  a  horizontal  plane. 

Since  the  form  of  the  earth  is  nearly  spherical,  the 
directions  of  gravity  at  difierent  points  will  converge 
towards  its  centre ;  but  the  length  of  the  earth's 
radius  is  so  great,  compared  with  the  dimensions  of 
the  bodies  usually  treated  of  in  mechanics,  that  we 
may  neglect  this  convergence,  and  consider  the 
directions  of  gravity,  for  all  the  points  of  the  same 
body,  as  parallel. 


STATICS.  61 

From  experiments  and  observations  which  will  be 
made  known  hereafter,  it  has  been  found, 

1"  That  the  intensity  of  gravity  at  the  surface  of 
the  earth,  though  always  the  same  at  the  same  place, 
varies  with  the  latitude  of  the  place ;  being  least  at 
the  equator,  and  increasing  as  we  approach  the  poles, 
the  increments  being  in  the  ratio  of  the  square  of 
the  sine  of  the  latittide. 

2"  That  its  intensity  varies  from  one  point  to 
another  of  the  same  vertical ;  diminishing  as  the 
distance  of  the  point  from  the  centre  of  the  earth 
increases,  in  the  ratio  of  the  square  of  the  distance. 

But  the  variations  of  intensity  from  these  two 
causes,  for  small  changes  of  distance,  are  so  minute 
that  the  action  of  gravity  upon  all  the  particles  of  a 
body  of  ordinary  size  may  be  considered  as  equally 
intense.  Since  then  within  the  requisite  limits, 
gravity  may  be  considered  as  constant  in  both  inten- 
sity and  direction,  we  may  regard  the  particles  of  a 
heavy  body  as  the  points  of  application  of  a  system 
of  equal  and  parallel  forces,  acting  vertically  and  in 
the  same  sense.  The  resultant  of  these  forces,  which 
is  equal  to  their  sum,  and  presses  the  body  vertically 
downwards,  will  evidently  constitute  what  is  called 
the  weight  of  the  body.  If  then  we  denote  the  weight 
of  the  body  by  W ;  the  number  of  material  particles 
composing  it,  by  M ;  and  the  effect  of  gravity  upon 
a  single  particle,  by  g,  we  shall  have 

W  =:  Mg [22] 


62  MECHANICS. 

An  immediate  inference  from  this  equation  is,  that 
in  homogeneous  bodies,  the  weights  are  proportional 
to  the  volumes ;  a  deduction  constantly  verified  by 
experiment. 

In  heterogeneous  bodies,  this  relation  between  the 
weights  and  volumes  does  not  hold  ;  the  weights  of 
equal  volumes,  when  compared,  being  found  unequal. 
Thus  a  cubic  inch  of  gold  is  found  to  weigh  about 
nineteen  times  as  much  as  a  cubic  inch  of  distilled 
water  at  a  certain  standard  temperature  ;  a  cubic 
inch  of  silver,  eleven  times  as  much. 

Since  the  weights  of  equal  volumes  of  two  sub- 
stances must  be  directly  as  the  numbers  of  material 
particles  which  they  contain,  equal  volumes  of  gold 
and  silver  must  contain,  the  one  nineteen,  the  other 
eleven  times  as  many  material  particles  as  the  same 
volume  of  distilled  water.  Thus  the  numbers  19 
and  11  express  the  relative  numbers  of  material 
particles,  or  the  relative  quantities  of  matter  con- 
tained in  equal  volumes  of  gold  and  silver;  the 
number  of  particles,  or  the  quantity  of  matter,  in  an 
equal  volume  of  water,  being  assumed  as  the  unit. 
The  numbers  which  thus  express  the  relative  quan- 
tities of  matter  referred  to  a  common  unit,  contained 
in  equal  volumes  of  heterogeneous  bodies,  are  called 
the  densities  of  these  bodies ;  thus  19  and  11  are  the 
densities  of  gold  and  silver  resjDCctively,  the  density 
of  distilled  water  being  taken  for  the  unit.  If  now 
we  denote  the  density  of  a  body  hy  D ;  its  volume, 
expressed  in  terms  of  the  unit  of  volume,  by  V ;  and 


STATICS.  63 

the  number  of  particles  thg^t  it  contains,  that  is,  its 

quantity  of  matter,  or  mass,  by  M,  we  shall  evidently 

have 

M  —    VD [ia] 

If  we  substitute  this  value  of  M  in  equation  [22], 

we  shall  get 

W  —    VDg [-24] 

Since  the  value  of  g,  the  general  symbol  denoting 
the  intensity  of  gravity,  varies  with  the  position  of 
the  place,  we  take  its  value  at  some  determinate 
place  for  the  unit  of  intensity,  and  consider  its 
general  value  as  expressed  in  terms  of  this  unit. 
We  thus  have  for  the  place  at  which  g-  =  1, 

W  =    VD; 

and  for  any  place  whatever, 

W  —    VDg, 

g  being  in  this  equation  the  ratio  of  the  intensities 
of  gravity  at  the  two  places.  If  the  unit  of  volume 
be  the  cubic  inch,  the  unit  of  weight  is  evidently  the 
weight  of  a  cubic  inch  of  distilled  water  at  the  place 
where  g  =  1,  and  W  designates  the  number  of  these 
units  which  the  weight  of  the  body  contains. 

42.  Since  the  weight  of  a  body  is  the  resultant  of 
a  system  of  parallel  forces  applied  at  points  of  which 
the  relative  positions  are  fixed,  it  must  have  a  de- 
terminate point  of  application.  This  point,  which  is 
the  centre  of  parallel  forces  for  the  case  in  question, 
is  called  the  centre  of  gravity  of  the  body.  The  posi- 
tion of  the  centre  of  gravity,  with  respect  to  thf 


64  MECHANICS. 

points  of  the  body,  is  invariable  ;  remaining  the 
same,  while  we  suppose  the  body  to  take  all  possible 
positions  in  space ;  for  a  change  in  the  position  of 
the  body  is  merely  equivalent  to  a  revolution  of  the 
directions  of  the  forces  about  their  points  of  applica- 
tion, an  operation  which  [Art.  25]  does  not  affect  the 
230sition  of  the  centre  of  parallel  forces. 

When  the  centre  of  gravity  of  a  body  is  supported, 
the  body  will  be  in  equilibrium  in  all  positions  about 
that  point ;  for,  in  all  positions,  the  dii^ction  of  the 
weight  of  the  body  will  pass  through  the  point  of 
support. 

43.  Conceive  now  a  body  to  be  divided  into  any 
number  of  parts,  and  suppose  the  weights  and  centres 
of  gravity  of  the  several  parts  to  be  known.  Let  the 
weights  of  the  parts  be  denoted  by  w,  w\  w",  etc.,  and 
the  co-ordinates  of  their  centres  of  gravity  by  x,  y  and 
z ;  x',  y  and  z' ;  x'\  y"  and  z'\  etc.  respectively ;  and 
let  the  w^eight  of  the  entire  body  be  denoted  by  JV, 
and  the  co-ordinates  of  its  centre  of  gravity  by  a: ,  y^ 
and  z^.  Then  since  W  is  the  resultant  of  the  parallel 
forces  w,  w\  w'\  etc.  applied  at  the  respective  centres 
of  gravity,  we  shall  have  [Art.  26]  to  determine  the 
co-ordinates  x^,  y^,  z^,  the  equations 

wx  +  w'x^  +  w"x''  -r  etc. 


X,    1= 


W 


ivi)  +  w'y'  4-  iv'hj"  -f  etc. 

—  JfT^  . 

icz  -f  xo'z^  -f  tf?"2"  +  etc. 


[25] 


STATICS.  65 

When  the  centres  of  gravity  of  the  several  parts 
are  situated  in  the  same  plane,  the  plane  of  xy  for 
example,  the  centre  of  gravity  of  the  entire  body 
will  be  found  in  that  plane  :  when  they  are  situated 
in  the  same  straight  line,  as  the  axis  of  x,  it  will  be 
found  in  that  line  [Art.  28]. 

The  above  equations  are  evidently  true,  whatever 
the  number  of  parts  into  which  we  suppose  the  body 
divided. 

44.  If  we  denote  the  masses  corresponding  to  the 
weights  W,  w,  w',  w'\  etc.  by  M,  m,  m',  m",  etc.,  we 
shall  have 

W  =  3Ig,     w  =  mg,     w'  =  m'g,     w''  =  7n"g,  etc. ; 

and  by  substituting  these  values  in  the  above  equa- 
tions, and  omitting  the  common  factor  g,  we  shall  get 

mx  +  m^x'  -\-  m'^x"  -f  etc. 


^'  -  M 

my  -\-  m'y'  +  7n"y"  +  etc. 

mz  +  m'z'  +  m"z"  +  etc. 


z,  = 


M 


[26] 


From  these  equations,  it  appears  that  the  position  of 
the  centre  of  gravity  is  independent  of  the  intensity 
of  gravity. 

45.  If  we  suppose  the  body  to  be  homogeneous, 
and  denote  its  volume  and  density  by  V  and  D 
respectively,  and  the  volumes  of  its  several  parts 
by  V,  v,  v",  etc.,  we  shall  have 


66  MECH.VNICS. 

M  =    VD,     m  =  vD,     m'  —  v'D,     m     —  v'D; 
and  by  substituting  these  values  in  equations  [26], 
and  omittino;  the  common  factor  B,  we  shall  get 


vx  +  v'x'  +  v"x"  +  etc. 

X,    Z=Z    Z-. 


vy  -\-  v'y'  +  v"y"  +  etc. 

vz  +  v'z'  -\-  v"z"  +  etc. 


[27] 


If  we  suppose  the  parts  into  which  the  body  is 
conceived  to  be  divided,  to  be  infinitely  small,  we 
may  derive  from  equations  [26]  and  [27]  the  follow- 
ing theorem  :  Th&sum  of  the  products  obtained  hy  multi- 
plying either  the  masses  or  the  volumes  of  the  elements^  or 
infinitely  small  parts  of  a  body,  by  their  respective  distan- 
ces from  any  plane  whatever,  is  equal  to  the  product  of  the 
entire  mass  or  volume  by  the  distance  of  its  centre  of  gravity 
from  the  same  plane. 

If  the  elements  of  the  body  are  all  situated  in  the 
same  plane,  or,  in  other  words,  if  the  body  itself  is  a 
material  plane,  we  may  consider  the  plane  with 
respect  to  which  the  moments  are  taken  as  reduced 
to  its  line  of  intersection  with  the  plane  of  the  ele- 
ments (the  two  planes  being  supposed  to  be  perpen- 
dicular to  each  other),  and  the  moments  are  then 
taken  with  respect  to  this  line. 

In  the  still  simpler  case  in  which  the  elements  are 
situated  in  the  same  straight  line,  we  may  consider 
the  line  to  which  the  moments  are  referred  as  reduced 
to  its  point  of  intersection  with  the  line  of  the  ele- 


STATICS.  67 

ments  (the  two  lines  being  supposed  perpendicular 
to  each  other),  and  the  moments  are  then  taken  with 
respect  to  this  point. 

It  will  be  recollected  that  these  products,  or 
moments  of  the  masses  or  volumes,  as  they  may  be 
called,  must  be  affected  with  opposite  signs,  accord- 
ing as  the  perpendiculars  are  situated  on  the  same 
or  opposite  sides  of  the  plane,  line  or  point. 

It  is  obvious,  that  if  in  any  case  the  sum  of  the 
moments  of  the  masses  or  volumes,  with  respect  to 
the  plane,  line  or  point,  is  equal  to  zero,  the  centre 
of  gravity  of  the  entire  body  must  be  situated  in 
the  j)lane,  line,  or  point. 

46 .  Determination  of  the  centres  of  gravity  of  particular 
bodies. 

The  application  of  the  preceding  equations  to  the 
determination  of  the  centres  of  gravity  of  particular 
bodies,  requires  in  general  the  use  of  the  integral 
calculus.  In  very  many  cases,  however,  these  points 
can  be  determined  by  the  most  elementary  processes. 
The  examples  which  follow  will  illustrate  the  me- 
thods commonly  employed. 

I.   Of  lines  and  surfaces. 

We  shall  first  consider  some  of  the  simplest  cases 
of  material  or  phj^sical  lines  and  surfaces  :  the  lines 
being  conceived  to  consist  of  single  series  of  mate- 
rial particles ;  the  surfaces,  of  single  laminae  of  parti- 
cles, the  particles  being  supposed  in  both  cases  to  be 
uniformlv  distributed. 


68  MECHANICS. 

1°  The  straight  line. 

The  centre  of  gravity  of  a  material  straight  line  is  its 
middle  point.  For,  drawing  through  this  point  any 
line  or  axis  whatever,  the  sum  of  the  moments  of  the 
particles  on  one  side  of  the  axis  is  obviously  equal 
to  the  sum  of  the  moments  of  the  particles  on  the 
opposite  side ;  and  these  sums  are,  moreover,  affected 
with  opposite  signs.  Their  algebraic  sum  is  therefore 
equal  to  zero,  and  hence  the  centre  of  gravity  of  the 
line  must,  by  the  last  paragraph  of  the  preceding 
article,  be  in  the  axis ;  but  it  is  also  in  the  line  itself, 
consequently  it  must  be  at  the  point  of  intersection 
of  the  line  and  the  axis. 

The  same  result  may  be  found  more  directly  by 
applying  the  ordinary  rule  for  parallel  forces.  Thus, 
let  the  weights  of  the  particles  be  regarded  as  a 
system  of  equal  and  parallel  forces  acting  in  pairs, 
the  components  of  each  pair  being  applied  on  oppo- 
site sides  of  the  middle  point  and  at  equal  distances 
from  it,  and  let  the  point  of  application  of  the 
resultant  of  the  whole  system  be  sought.  The  re- 
sultant of  each  pair,  and  hence  the  resultant  of  the 
whole  system,  will  be  found  to  pass  through  the 
middle  point  of  the  line. 

2"   The  perimeter  of  a  polygon. 

To  determine  the  centre  of  gravity  of  the  peri- 
meter of  a  polygon  :  in  the  plane  of  the  polygon, 
draw  the  co-ordinate  axes  OX  and  OY,  and  deter- 
mine the  co-ordinates  of  the  centres  of  gravity  of  the 
several  sides  referred  to  these  axes.   Then  in  the 


STATICS.  69 

first  two  of  equations  [27]  Art.  45,  substitute  for  the 
volumes  v,  v',  v'\  etc.  the  lengths  of  the  sides ;  for  x 
and  y,  x  and  y\  x"  and  y" ,  etc.  the  co-ordinates  of 
their  centres  of  gravity  respectively ;  and  for  the 
volume  F,  the  entire  perimeter  of  the  polygon.  The 
values  of  x^  and  v/ ,  thus  determined,  will  be  the 
values  of  the  co-ordinates  of  the  centre  of  gravity 
of  the  polygon. 

Another  method  consists  in  regarding  the  weights 
of  the  several  sides,  applied  at  the  respective  centres 
of  gravity,  as  a  system  of  parallel  forces,  and  deter- 
mining the  point  of  application  of  the  resultant  by 
the  ordinary  rule. 

3°  The  arc  of  a  circle. 

Let  AFB  [Fig.  28]  be  an  arc  of  a  circle,  and  let 
MN  be  one  of  the  infinitely  small  parts  or  particles 
of  which  we  suppose  it  made  up.  Draw  the  diame- 
ter LR  parallel  to  the  chord  AB ;  and  from  Q  the 
middle  point  of  MN,  draw  QP  perpendicular  to  LR. 
Draw  also  the  radius  QO,  the  line  MI  parallel  to  AB, 
and  the  lines  MH  and  NK  perpendicular  to  AB. 
The  similar  triangles  MNI  and  QOP  give 

MN    :    MI    ::    QO    :    QP, 
or  MN  X  QP  =  MI  X  QO 

=  HK  X  QO. 

The  first  member  of  this  equation  is  the  moment  of 
MN  with  respect  to  the  diameter  LR  ;  and  the  second 
is  the  product  of  the  projection  of  MN  on  the  chord 
AB,  by  the  radius  of  the  circle.  Thus  the  moment 
of  each  particle  of  the  arc  is  equal  to  the  product 


70  MECHANICS. 

of  its  projection  on  the  chord,  by  the  radius  of  the 
circle  :  consequently  the  sum  of  the  moments  of  all 
the  particles  of  the  arc  with  respect  to  LR,  is  equal 
to  the  product  of  the  chord  by  the  radius,  or  to 
x\B  X  OQ.  But  the  sum  of  the  moments  of  the  par- 
ticles of  the  arc,  with  respect  to  the  radius  OF  drawn 
to  its  middle  point,  is  evidently  equal  to  zero,  and 
hence  OF  must  contain  the  centre  of  gravity  of  the 
arc.  If  then  we  suppose  the  centre  of  gravity  to  be 
at  C,  the  expression  of  the  moment  of  the  arc  with 
respect  to  LR  will  be,  arc  AFBxCO;  and  hence 
[Art.  45]  we  shall  have 

arc  AFB  X  CO  =  AB  X  OQ, 
or  arc  AFB     :     AB     : :     OQ     :    CO. 

Hence  the  centre  of  gravity  of  the  arc  of  a  circle  is  on  the 
radius  which  bisects  the  arc,  at  a  point,  the  distance  of  which 
from  the  centre  of  the  circle  is  a  fourth  proportional  to  the 
length  of  the  arc,  its  chord,  and  radius. 

4"  The  area  of  a  parallelogram. 

The  centre  of  gravity  of  the  area  of  a  parallelogram  is 
at  the  intersection  of  its  diagonals.  For  the  sum  of  the 
ifioments  of  the  particles  of  the  parallelogram,  with 
respect  to  each  of  its  diagonals,  is  evidently  equal 
to  zero,  and  hence  each  diagonal  must  pass  through 
its  centre  of  gravity ;  consequently  the  centre  of 
gravity  of  the  parallelogram  must  be  at  the  inter- 
section of  these  lines. 

Another  method  of  determining  the  centres  of 
gravity  of  plane  figures,  consists  in  regarding  them 


STATICS.  71 

as  made  up  of  physical  lines.  To  apply  this  to  the 
case  of  the  parallelogram,  conceive  it  made  up  of 
lines  parallel  to  one  of  its  diagonals ;  the  other  dia- 
gonal bisecting  these  parallels,  and  hence  passing 
through  their  centres  of  gravity,  will  contain  the 
centre  of  gravity  of  the  entire  figure  ;  but  this  pro- 
perty is  common  to  the  two  diagonals  :  consequently 
the  centre  of  gravity  of  the  parallelogram  must  be 
at  their  intersection. 

5°  The  area  of  a  triangle. 

To  find  the  centre  of  gravity  of  a  triangle  ABD 
[Fig.  29] :  From  the  vertices  D  and  B  draw  the  lines 
DE  and  BF  to  the  middle  points  of  the  opposite  sides 
AB  and  AD,  and  join  the  points  E  and  F ;  the  centre 
of  gravity  will  be  at  C.  For,  regarding  the  triangle 
as  made  up  of  physical  lines  parallel  to  AB,  its 
centre  of  gravity  must  be  in  the  line  DE  which 
bisects  these  parallels  :  it  must  also  be  in  BF,  which 
bisects  the  lines  drawn  parallel  to  AD ;  consequently 
it  is  at  the  point  C. 

To  determine  the  position  of  this  point  we  have, 
from  the  similar  triangles  BCD  and  FCE,  ABD  and 
AEF, 

CD 


hence 


and  CE 


CE    ::    BD    : 

EF 

::    AB   : 

AE 

::       2   : 

1 

2CE  =  CD, 

3CE  =  ED, 

ED 

72  MECHANICS. 

Hence  the  centre  of  gravity  of  the  area  of  a  triangle  is 
0 1  the  straight  line  drawn  from  any  one  of  its  vertices,  to 
the  middle  point  of  the  opposite  side,  at  a  distance  from 
this  point  equal  to  one-third  of  the  length  of  the  line. 

6"   The  area  of  a  polygon. 

To  determine  the  centre  of  gravity  of  the  area  of 
a  polygon,  divide  the  polygon  into  triangles,  and 
find  the  centre  of  gravity  of  each  triangle  by  the 
preceding  method ;  then  proceed-  as  in  the  case  of 
the  perimeter. 

7°   The  area  of  a  circular  sector. 

Let  AOB  [Fig.  30]  be  a  circular  sector,  and  let  it 
be  supposed  to  be  divided  into  an  infinite  number  of 
infinitely  small  equal  triangles,  having  their  bases 
in  AB,  and  their  vertices  at  the  centre  0.  The 
centre  of  gravity  of  each  of  these  triangles  will  be 
in  the  radius  drawn  to  the  middle  of  its  base,  at  a 
distance  from  the  centre  of  the  circle  equal  to  two- 
thirds  of  the  length  of  the  radius  :  hence  the  centre 
of  gravity  of  the  sector  will  be  the  same  as  that  of 
the  arc  FG  described  with  a  radius  01,  equal  to  two- 
thirds  the  radius  of  the  sector.  It  will  therefore 
[No.  3"  of  this  Art.]  l^e  on  the  radius  OD  drawn  to 
the  middle  of  the  arc  AB,  at  a  point  C,  the  distance 
of  which  from  0  will  be  given  by  the  proportion 

arc  FIG     :    FG    : :    01      :    OC ; 
or  arcADB     :     AB    : :     fOD     :     OC. 

Hence  the  centre  of  gravity  of  the  area  of  a  circular 
sector  is  on  the  radius  ivhich  bisects  the  arc  of  the  sector, 


STATICS.  73 

at  a  point,  the  distance  of  wliichfrom  the  centre  is  a  fourth 
proportional  to  the  arc,  its  chord,  and  tivo-thirds  of  the 
radius. 

8"  The  surface  of  a  spherical  zone. 

The  centre  of  gravity  of  the  surface  of  a  spherical  zone 
is  at  the  middle  point  of  its  axis,  or  the  line  joining  the 
centres  of  its  bases.  For,  conceive  the  entire  zone 
divided  into  an  infinite  number  of  infinitely  small 
zones  of  the  same  altitude,  by  planes  parallel  to  its 
bases ;  the  centres  of  gravity  of  these  infinitesimal 
zones  will  be  on  the  axis  of  the  entire  zone  ;  but 
these  zones  are  equivalent  in  surface  :  hence  their 
common  centre  of  gravity,  or  the  centre  of  gravity 
of  the  entire  zone,  will  be  at  the  middle  point  of 
the  axis. 

II.  Of  solids. 

1°  The  parallelopipedon. 

Let  AH  [Fig.  31]  be  a  parallelopipedon.  Through 
the  opposite  edges  BG  and  EI,  AF  and  DH,  let  the 
planes  BGIE,  ADHF  be  drawn ;  and  through  C  the 
middle  point  of  00'  the  intersection  of  these  planes, 
let  the  plane  A'B'D'E'  be  drawn  at  right  angles  to 
00'. 

The  matter  of  the  solid  is  obviously  so  disposed 

on  the  ojDposite  sides  of  these  planes,  that  the  sum 

of  the  moments  of  its  particles  with  respect  to  each 

plane  is  equal  to  zero  :  consequently  the  centre  of 

gravity  of  the  parallelopipedon  is  at  C  the  common 

point  of  intersection  of  the  three  planes.  This  point 

is  obviously  the  middle  point  of  any  one  of  the 

diagonals  of  the  solid. 

10 


74  MECHANICS. 

Another  method  of  determining  the  centre  of 
gravity  of  solids,  consists  in  regarding  them  as  made 
up  of  physical  j^lanes.  Thus,  in  the  case  just  consi- 
dered, we  may  suppose  the  solid  made  up  of  planes 
parallel  to  its  base  ABDE  :  the  ce?itre  of  gravity  ivill 
obviously  he  at  C  the  middle  of  the  line  00'  which  joins 
the  centres  of  gravity  of  the  two  bases. 

2"  The  pijramid. 

Let  A-BED  [Fig.  32]  be  a  triangular  j^yramid. 
From  the  vertices  A  and  D,  draw  to  F  the  middle 
point  of  the  edge  BE,  the  straight  lines  AF  and  DF  ; 
take  FK  equal  to  one-third  of  AF,  and  FH  equal  to 
one-third  of  FD,  and  draw  DK  and  AH  :  the  point  C 
in  which  DK  and  AH  intersect,  is  the  centre  of 
gravity  of  the  pyramid.  For,  conceive  the  pyramid 
to  be  made  up  of  planes  parallel  to  the  face  BED  :  the 
line  AH  will  pass  through  the  centres  of  gravity  of 
all  these  planes,  and  will  therefore  contain  the  centre 
of  gravity  of  the  p^'ramid.  In  like  manner  it  may 
be  shown  that  the  line  DK  will  also  contain  this 
point ;  consequently  it  must  be  at  C  the  point  of 
intersection  of  these  lines.  To  determine  its  position, 
joining  the  points  H  and  K,  we  have,  from  the 
similar  triangles  ACD,  HCK,  AFD,  KFH, 
CH 


hence 


and 


CA    ::    HK 

:    DA 

::    FH 

:    FD 

::       1 

:       3 

3  CH  =  CA, 

4CH  =  AH, 

STATICS.  75 

Hence  the  centre  of  gravity  of  a  triangular  pyramid  is 
on  the  straight  line  drawn  from  the  vertex  of  any  one  of 
its  angles,  to  the  centre  of  gravity  of  the  opposite  face,  and 
at  a  distance  from  the  face  equal  to  one  fourth  of  the  length 
of  the  line. 

It  can  now  be  easily  shown  that  the  centre  of  gravity 
of  any  pyramid  ivhatevcr  is  on  the  straight  line  drawn  from 
its  vertex  to  the  centre  of  gravity  of  its  base,  at  a  distance 
from  the  base  equal  to  one  fourth  the  le7igth  of  the  line. 

Let  A-BDE....  [Fig.  33]  be  the  pyramid.  From 
the  vertex  A,  draw  the  straight  line  AI  to  I  the  cen- 
tre of  gravity  of  the  base ;  and  divide  the  base  into 
triangles,  by  joining  its  vertices  with  the  point  I. 
From  the  centres  of  gravity  of  the  triangles,  draw 
lines  to  the  vertex  ;  and  taking  IC  equal  to  one- 
fourth  of  AI,  through  C  draw  a  plane  parallel  to  the 
base  of  the  polj^gon.  The  point  C  will  be  the  centre 
of  gravity  of  the   pyramid.   For,  since   the  plane 
drawn  through  C  cuts  all  the  lines  drawn  from  A  to 
the  base  proportionally,  it  will  contain  the  centres  of 
gravity  of  the  triangular  pyramids  A-IBD,  A-IDE, 
etc.,  and  consequently  will  also  contain  the  centre  of 
gravity  of  the  entire  pyramid  ;  and  since  the  straight 
line  AI  passes  through  the  centres  of  gravity  of  all 
the  planes  drawn  parallel  to  the  base,  it  also  will 
contain  the  centre  of  gravity  of  the  entire  pyramid  : 
therefore  the  centre  of  gravity  of  the  entire  pyramid 
will  be  at  the  intersection  of  the  line  and  plane. 

The  result  just  obtained,  being  independent  of  the 
number  of  sides  of  the  base  of  the  pyramid,  is  true 


76  MECHANICS. 

when  the  number  becomes  infinite,  and  the  base  a 
re-entering  curve  :  it  is  therefore  true  for  a  cone  of 
any  base  whatever. 

Any  polyhedron  being  divisible  into  pyramids,  its 
centre  of  gravity  may  be  found  by  conceiving  it  to 
be  thus  divided,  and  determining  the  centre  of 
gravity  of  each  pyramid  thus  formed,  with  respect 
to  three  co-ordinate  planes,  and  then  applying  equa- 
tions [27]  of  article  45. 

3"  The  spherical  sector. 

The  centre  of  gravity  of  a  spherical  sector,  or  the  solid 
generated  hy  the  revolution  of  a  circular  sector  about  one 
of  its  sides,  is  on  the  axis  of  the  sector,  at  a  distance  from 
the  centre  of  the  sphere  equal  to  three  fourths  of  the  radius., 
minus  three-eighths  of  the  altitude  or  axis  of  the  corre- 
sponding spherical  calotte.^ 

For  conceive  the  entire  sector  to  be  made  up  of 
an  infinite  number  of  infinitely  small  equivalent 
pyramids,  having  their  vertices  at  the  centre  of  the 
sphere.  The  centre  of  gravity  of  each  of  the  pj'ra- 
mids  will  be  on  the  radius  drawn  to  the  centre  of 
gravity  of  its  base,  at  a  distance  from  the  centre  of 
the  sphere  equal  to  three-fourths  of  the  radius.  The 
centre  of  gravity  of  the  entire  sector  will  therefore 
be  the  same  as  that  of  a  spherical  calotte  concentric 
with  the  calotte  of  the  sector,  and  having  its  radius 
equal  to  three-fourths  the  radius  of  the  sphere  :  hence 
the  centre  of  gravity  of  the  entire  sector  will  be  at 

*"  Spherical  ca.lotle,'*^  or  "cap,"  a  spherical   zone  of  one  base. 


STATICS.  77 

the  middle  point  of  the  axis  of  this  concentric  calotte. 
If  we  denote  the  height  of  the  calotte  of  the  sector 
by  h,  the  height  of  the  concentric  calotte  will  be  %h : 
hence,  denoting  the  radius  of  the  sphere  by  r,  the 
distance  of  the  centre  of  gravity  of  the  entire  sector 
from  the  centre  of  the  sphere  will  be  fr— |/i. 


MACHINES. 

47.  A  machine  is  an  instrument  by  means  of  which 
a  force  may  be  made  to  act  upon  points  that  lie 
without  its  direction. 

The  simple  machines,  of  which  all  others  are  com- 
posed, are  the  cord  or  rope  machine,  the  lever,  and  the 
inclined  plane.  Certain  modifications  or  combinations 
of  these  are  frequently  ranked  among  the  simple 
machines,  viz.  the  pulley,  the  loheel  and  axle,  the  screw, 
and  the  wedge. 

The  force  employed  in  working  a  machine,  is 
called  the  poiver ;  and  the  force  to  be  overcome,  the 
resista7ice. 

In  discussing  the  theory  of  machines,  we  seek  only 
the  conditions  of  equilibrium  of  the  power  and  re- 
sistance. The  consideration  of  the  motion  which 
ensues  when  the  power  is  increased  beyond  what  is 
required  for  an  equilibrium,  belongs  to  dynamics. 

In  the  first  investigations,  we  omit  the  considera- 
tion of  the  weight  of  the  machine,  the  stiffiiess  of 
cords,  the  flexibility  of  rods,  friction,  etc.  etc. 


78  MECHANICS. 

THE    EOPE    MACHINE. 

48.  The  simplest  form  of  the  rope  machme  is  that 
represented  in  figure  34,  in  which  three  cords  AB, 
AC  and  AD,  lying  in  the  same  plane,  are  firmly 
united  at  a  point  A ;  and  the  forces  P,  Q  and  R  are 
applied  at  their  extremities  B,  C  and  D.  The  condi- 
tions of  equilibrium,  in  this  case,  are  evidently 
expressed  by  the  proportion  [Art.  12], 

P    :     Q     :     R     ::     s'mp     :     siu^     :     sinr; [28] 

p,  q  and  r  denoting  the  angles  made  by  the  cords,  or 
the  directions  of  the  forces  as  represented  in  the 
figure. 

49.  If  the  forces  P  and  Q  be  suppressed,  and  the 
extremities  B  and  C  be  attached  to  fixed  points,  the 
conditions  of  equilibrium  will  still  be  given  by  pro- 
portion [28] ;  and  P  and  Q  will,  in  this  case,  denote 
the  re-actions  of  the  points  respectively. 

50.  When  the  extremities  B  and  C  are  fixed,  and 
the  cord  AD  is  attached  to  a  ring  which  is  capable 
of  moving  freely  upon  BAG,  it  is  evident,  that  as  the 
ring  slides  along  this  line,  the  point  A  will  describe 
an  ellipse.  Consequently  in  order  to  an  equilibrium, 
the  direction  of  AD  must  coincide  with  the  normal 
to  the  curve  at  the  point  A,  and  hence  must  bisect 
the  angle  BAC  of  the  radii  vectores.  We  shall  then 
have 

|)  z=  g  and  P  ==    Q, 
and  P     :     R     ::     sinp     :     sin  r. 


STATICS.  79 

51.  The  common  intensity  of  two  equal  and  oppo- 
site forces  applied  at  the  extremities  of  a  cord,  in  the 
direction  of  its  length,  so  as  to  extend  it,  is  called  the 
tension  of  the  cord ;  thus,  P,  Q  and  R  are  the  tensions 
of  the  cords  AB,  AC  and  AD  respectivelj^  The 
effect  upon  the  cord  will  obviously  be  the  same, 
whether  the  two  equal  forces  thus  act  at  its  extremi- 
ties, or  one  of  them  be  suppressed,  and  the  extremity 
to  which  it  was  applied  be  attached  to  a  fixed  point. 

52.  The  case  in  which  there  is  anj^  number  of 
cords  united  at  the  same  point,  and  solicited  by 
forces  lying  in  the  same  plane,  is  immediately  redu- 
cible to  that  just  considered. 

53.  Another  form  of  this  machine,  called  the  funi- 
cular polygon^  is  represented  in  figure  [35];  in  which 
EABCF,  AG,  BH,  CI,  are  supposed  to  be  cords 
united  at  the  points  A,  B  and  C,  and  solicited  at 
their  extremities  by  the  forces  P,  P\  P",  etc.  acting 
in  the  same  plane.  When  these  forces  are  in  equili- 
brium with  each  other,  it  can  readily  be  shown  : 

1"  That  the  forces  must  be  such  that  they  will 
be  in  equilibrium  when  applied  at  a  single  point  (C, 
for  example,  as  represented  in  the  figure),  parallel  to 
their  primitive  directions,  and  hence  that  the  condi- 
tions of  equilibrium  are  the  same  as  those  of  Art. 
[16]. 

2°  Denoting  the  tensions  of  the  portions  AB  and 
BC  of  the  cord  EABCF  by  S  and  Q,  and  the  angles 
at  A,  B  and  C  by  s  and  p,  q  and  s\  p'^  and  q,  as  repre- 
sented in  the  figure,  and  applying  the  principle  of 


80  MECHANICS. 

proportion  [28],  that  tlie  relations  of  the  tensions  P 
and  P"'  will  be  given  by  the  proportion 

P     :     P^'     :  :     sin/;  X  sin  s'  X  sin  5'     I     sin  s  X  sin  q'  X  sinp'^. 

o"*.  That  when  the  forces  P\  P",  P  ",  are  parallel 
to  each  other  [Fig.  36],  and  hence  the  sums  p  -f-  q\ 
s'  +  p'^,  are  each  equal  to  180^, 

P     :     P''     :  :     sin  q     :     sin  s. 

4".  That  when  the  cord  EBF  [Fig.  37],  sustained 
by  the  forces  P  and  P"'  applied  at  E  and  F,  is  sub- 
jected to  no  other  additional  force  than  gravity, 

P     :    P'^    :    R     ::    sinFMa    :    sinEMG     :    sinEMF; 
R  denoting  the  weight  of  the  cord,  and  EM  and  PM 
being  tangents  to  the  cord  at  the  points  E  and  F, 
and  MG  a  vertical  line  drawn  through  their  point 
of  meeting. 

THE    LEVER. 

54.  A  lever  is  a  bar  or  rod  of  any  form  whatever, 
capable  of  motion  about  a  fixed  axis  or  support 
called  the  fulcrum.  In  investigating  the  properties 
of  the  lever,  we  conceive  it  reduced  to  a  material 
line. 

Let  the  line  ACDB  [Fig.  38]  represent  a  lever,  and 
F  its  fulcrum  ;  and  suppose  it  to  be  acted  upon  by 
the  forces  P,  P ,  etc.  tending  to  turn  it  in  one  sense 
about  the  point  F,  and  P",  P",  etc.  tending  to  turn 
it  in  the  opposite  sense  about  this  point,  all  the  forces 
being  supposed  to  act  in  the  same  plane.  Also  let 
the  lines  p,  p ,  etc.,  p'\  p"\  etc.  be  drawn  from  F 


STATICS.  81 

perpendicular  to  tlie  directions  of  the  forces.  Then 
the  conditions  of  equilibrium  of  the  forces  will  be 
expressed  by  thie  equation  [Art.  35] 

Pp  +  P'p'  +  etc.   =  P"p"  +  P"'p"'  -f  etc. 

In  this  equation,  the  forces  P,  P\  etc.  may  be  sup- 
posed to  represent  the  powers  applied  to  the  lever, 
and  P",  P",  etc.  the  resistances  to  be  overcome. 
The  weight  of  the  lever  may  be  regarded  as  a  verti- 
cal force  applied  at  its  centre  of  gravity. 

55.  The  case  most  frequently  occurring  in  practice, 
is  that  in  which  only  two  forces  act  upon  the  lever. 
The  preceding  equation  is  then  reduced  to 

Pp  —  P"p", 
and  we  have 

P    :    P"    ::    p"    :    p. 

Thus,  in  this  case,  the  power  and  the  resistance  are 
inversely  as  the  perpendiculars  drawn  from  the  fulcrum 
to  their  directions. 

56.  When  the  lever  AB  [Fig.  39]  is  straight,  and 
the  two  forces  act  in  parallel  directions,  we  get  from 
the  similar  triangles  FAM,  FBN, 

p"    :    P    ::    ¥B    :    FA, 
and  hence  we  have 

P    :    P"    ::    FB    :    FA; 
that  is,  the  power  and  the  resistance  inversely  as  the  dis- 
tances of  their  points  of  application  from  the  fulcrum. 
These  distances  are  called  the  arms  of  the  lever. 

11 


82  MECHANICS. 

57.  The  pressure  upon  the  fulcrum  is  evidently 
equal  to  the  resultant  of  all  the  forces  which  act 
upon  the  lever.  When  the  lever  is  not  retained  by 
a  fixed  axis,  but  only  rests  upon  a  fixed  support,  it 
is  essential  to  the  equilibrium  that  the  direction  of 
the  resultant  should  be  perpendicular  to  the  lever. 

58.  According  to  the  relative  positions  of  the 
power,  the  resistance  and  the  fulcrum,  levers  have 
been  divided  into  three  kinds.  A  lever  is  said  to  be 
of  the  1st  kind,  when  the  power  and  the  resistance 
are  applied  on  opposite  sides  of  the  fulcrum,  as  in 
Fig.  40 ;  of  the  2d  kind,  when  these  forces  are  applied 
on  the  same  side  of  the  fulcrum,  the  resistance  being 
the  nearer  to  the  fulcrum,  as  in  Fig.  41 ;  and  of  the 
3d  kind,  when  these  forces  are  on  the  same  side  of 
F,  and  the  power  is  the  nearer  to  the  fulcrum,  as  in 
Fig.  42. 

THE    INCLINED    PLANE. 

59.  This  machine  consists  essentially  of  a  plane, 
inclined  at  any  angle  whatever  to  a  horizontal  plane. 
Conceive  a  body  to  be  in  equilibrium  on  an  inclined 
plane  :  it  is  evident  that  the  force  P  which  prevents 
the  body  from  sliding  down  the  plane,  and  P'  the 
gravity  of  the  body,  must  have  a  resultant  perpen- 
dicular to  the  plane,  and  meeting  it  within  the  limits 
of  the  base  of  the  body.  The  plane  of  these  forces 
will  be  vertical,  and  also  perpendicular  to  the  incli- 
ned plane.  The  section  of  the  inclined  plane  and  the 
body,  by  the  plane  of  the  forces,  is  represented  in 


STATICS.  83 

Fig.  43,  in  which  AB  is  the  section  of  the  inclined 
plane,  and  MLN  that  of  the  body,  and  BC  and  AC  are 
horizontal  and  vertical  lines  meeting  each  other  at 
the  point  C ;  AB,  BC  and  AC  being  the  length,  base, 
and  height  of  the  plane  respectively.  Let  the  line 
KE  be  the  direction  of  the  force  P;  KF,  which  we 
suppose  to  pass  through  the  centre  of  gravity  of  the 
body,  that  of  P';  and  KG  perpendicular  to  AB,  that 
of  the  resultant  R.  Then  denotino;  the  anoies  as 
represented  in  the  figure,  we  shall  have 

P     :     P'     :     R     ::     s'lnp     :     sin//     :     sin?-; 

or,  since  p  is  equal  to  i  the  inclination  of  the  j^lane, 
and  sin  i  =  y  5 

P     :     P      :     R     ::     h     :     Is'mp'     :     Is'mr [29] 

l*".  When  the  direction  of  P  is  parallel  to  the  in- 
clined plane,  p'  =  90°,  and  we  get  from  the  above 

proportion, 

P    :    P    ::    h    :    I; 

that  is,  the  power  to  the  weight  of  the  body,  as  the  height 
of  the  plane  to  its  length. 

2°.  When  the  direction  of  P  is  parallel  to  the  base 
of  the  plane,  p'  ~  (90°—  ?'),  and  we  have 

P    :    P'    ::    h    :    icosi 
::   h    :    b; 

that  is,  the  power  to  the  weight,  as  the  height  of  the  plane 
to  its  base. 

We  have  regarded  the  plane  as  inclined  with  re- 
spect to  the  horizon,  and  supposed  the  force  P'  to  be 
gravity ;  but   the  above   results  will   obviously  be 


84  MECHANICS. 

true  if  we  consider  the  plane  as  inclined  with  respect 
to  any  assumed  plane,  and  substitute  for  gravity  any 
force  acting  perpendicular  to  the  latter  plane. 

THE    PULLEY. 

60.  The  pulley  is  a  small  grooved  wheel,  capable 
of  motion  about  an  axis,  and  having  its  circumfe- 
rence partly  enveloped  by  a  cord,  to  the  extremities 
of  which  the  forces  are  applied.  The  axis  is  sup- 
ported by  a  frame  called  the  block ;  and  according  as 
the  block  is  fixed  or  cajDable  of  motion,  the  pulley  is 
said  to  be  fixed  or  movable. 

61.  The  fixed  pulley. 

Let  IGHK  [Fig.  44]  represent  a  fixed  pulley,  and 
EIKHF  a  cord  enveloping  the  arc  IKH  of  its  circum- 
ference ;  and  suppose  the  two  forces  P  and  Q  to  be 
applied  at  the  points  E  and  F,  in  the  directions  IE 
and  HF  tangent  to  the  circumference  at  the  points 
I  and  H.  Produce  EI  and  FH,  and  let  the  points  of 
application  of  P  and  Q  be  transferred  to  A  the  point 
of  meeting  of  these  lines.  Then  since  0  is  the  only 
fixed  point  in  the  pulley,  in  order  to  an  equilibrium, 
the  direction  of  the  resultant  R  of  P  and  Q  must 
pass  through  that  point  :  hence  it  must  bisect  the 
angle  EAF,  and  consequently  the  forces  P  and  Q 
must  be  equal  to  each  other.  Let  the  resultant  be 
represented  by  AD,  and  complete  the  parallelogram 
ABDC  :  then  P  and  Q  will  be  represented  by  the 
equal  lines  AB  and  AC  respectively,  and  we  shall 
have 


STATICS.  85 

P    :    R    ::    AB    :    ATI. 
But  from  the  similar  triangles  ABD,  lOH,  we  get 

AB    :    AD    ::    10    :    IH; 
consequently  we  have 

P  :  R  ::  10  :  IR. 
Thus,  in  the  fixed  pulley,  each  of  the  forces  applied  to 
the  cord  or  rope  is  to  their  resultant,  or  the  pressure  upon 
the  point  of  support,  as  the  radius  of  the  pulley  is  to  the 
chord  of  the  arc  with  which  the  rope  is  in  contact.  The 
fixed  pulley  is  employed,  when  it  is  desired  to  change 
the  direction  of  a  force  without  affecting  its  intensity. 

62.  Movable  pulley. 

In  the  movable  pulley,  one  extremity  of  the  cord 
is  attached  to  a  fixed  point  F  [Fig.  45] ;  and  a  power 
P  applied  to  the  other  extremity,  holds  in  equili- 
brium a  force  R  applied  in  a  direction  passing 
through  the  centre  0.  The  resistance  R  is  usually 
the  weight  of  a  body  suspended  from  the  centre.  In 
place  of  the  re-action  of  the  fixed  point  F,  we  may 
substitute  a  force  Q,  and  consider  the  machine  as  in 
equilibrium  under  the  action  of  the  forces  P,  Q  and 
R,  applied  at  points  entirely  free.  The  condition  of 
equilibrium  will  obviously  be,  that  the  resultant  of 
P  and  Q  must  be  equal  and  contrary  to  R.  Hence  it 
may  be  readily  inferred  that  P  and  Q  are  equal,  and, 
as  in  the  preceding  case,  that 

P    :    R    ::    10    :    IR. 

Thus,  in  the  movable  pulley,  the  power  is  to  the 
resistance,  as  the  radius  of  the  pulley  is  to  the  chord  of 
the  arc  in  contact  with  the  rope. 


86 


MECHANICS. 


1°  When  the  cords  EI  and  FH  are  parallel,  the 
chord  IH  becomes  a  diameter,  and  we  have 
P    :    JR    ::    1    :    2; 

that  is,  the  resistance  equal  to  twice  the  power. 

2°  When  the  arc  IKH  in  contact  with  the  cord 
is  equal  to  60  ,  we  have 

P  —  R; 

and  when  this  arc  is  less  than  60°, 

P>jR. 

63.  Systems  of  pulleys. 

1°  Let  C,  C,  C",  be  a  system  of  movable  pulleys, 
connected  with  each  other  in  the  manner  represented 
in  figure  46,  in  which  F,  F',  F",  are  fixed  points,  and 
the  power  P  applied  in  the  direction  I"E  holds  in 
equilibrium  the  weight  R  suspended  from  C.  Denot- 
ing the  tensions  of  the  cords  IC,  I  C"  by  P"  and  P', 
we  have 


p 

R     : 

:    IC 

IH, 

P' 

pn     . 

:   rc 

I'H', 

p 

P     : 

:    l"C" 

I"H' 

and  hence 

P    :    R    ::    ICxICxT'C"    :    IH  x  I'H' x  I"H". 

That  is,  in  this  system  of  pulleys,  the  power  is  to  the 
resistance,  as  the  product  of  the  radii  of  the  pulleys  is  to  the 
product  of  the  chords  of  the  arcs  in  contact  luith  the  ropes. 

When  the  cords  are  parallel,  as  in  figure  47,  the 
above  proportion  is  reduced  to 

P    :    R    ::    1    :    2\ 
Hence,  in  a  system  of  this  kind,  in  which  n  movable 


STATICS.  87 

pulleys  are  employed,  we  shall  have 
P    :    jR    ::    1    :    2«. 

By  means  of  movable  pulleys  arranged  in  this 
manner,  a  great  weight  may  be  moved  by  a  small 
force ;  but  it  will  readily  be  perceived  that  the  gain 
in  power  will  be  attended  w  ith  a  loss  in  respect  to 
time. 

2"  A  combination  more  convenient  than  the  pre- 
ceding, is  that  in  which  several  pulleys,  both  fixed 
and  movable,  are  embraced  by  a  single  cord  as  re- 
presented in  Fig.  48,  in  which  the  several  parts 
X,  y,  z,  etc.  of  the  cord  are  supposed  to  be  parallel 
to  each  other.  In  this  arrangement,  we  may  obviously 
consider  the  tensions  of  these  portions  of  the  cord 
as  a  system  of  equal  and  parallel  forces  acting  in 
the  same  sense,  of  which  the  resultant  is  equal  to 
the  weight  or  resistance  R.  Thus  in  the  case  of 
three  movable  pulleys,  in  which  the  cord  has  six 
branches,  we  have 

P    :    i2    ;:    1    :    6; 

and  hence  denoting  the  number  of  pulleys  by  ?z,  we 

have  generally 

P     \     R     \\     \     \     In. 

The  method  of  determining  the  j)ower  necessary 
to  put  in  equilibrium  the  weight  of  the  several 
movable  pulleys,  is  sufficiently  obvious. 

THE    WHEEL    AND    AXLE. 

64.  This  machine  consists  of  a  wheel  A  A'  [Fig.  49], 
and  a  cylinder  or  axle  BB'  so  adjusted  as  to  have  a 
common  axis,  and  firmly  connected  with  each  other. 


88  MECHANICS. 

The  cylinder  is  supported  at  its  extremities  in  such 
a  manner  as  to  admit  of  only  a  motion  of  rotation 
about  the  common  axis.  The  power  P  and  the 
resistance  R  are  applied  at  points  of  the  wheel  and 
axle,  respectively,  in  the  direction  of  tangents  to 
their  circumferences.  The  mode  of  operation  of  the 
machine  is  indicated  in  the  figure.  The  case  of  an 
equilibrium  between  P  and  R  in  this  machine  is 
obviously  comprehended  in  the  more  general  case 
already  considered  in  article  40.  Referring  then  to 
this  article,  and  denoting  the  radius  of  the  wheel 
by  p,  and  that  of  the  axle  by  r,  we  get 
Pp  —  Rr, 

or  P    :    R    ::    r    :    p. 

That  is,  in  the  equilibrium  of  the  wheel  and  axle, 
the  power  is  to  the  resistance  as  the  radius  of  the  axle  is 
to  the  radius  of  the  wheel. 

When  P  and  R  act  upon  the  machine  by  means 

of  cords,  we  must  suppose  these  forces  applied  to  the 

/  axes  of  the  cords,  and  increase  the  radii  of  both  the 

wheel  and  axle  by  the  radii  of  the  cords  respectively. 

65.  A  combination  of  wheels  and  axles  is  some- 
times used.  One  of  these  compound  machines,  in 
which  three  of  the  simj)ler  machines  are  employed, 
is  represented  in  figure  50.  To  find  the  relation  in 
this  case  between  the  power  P  and  the  resistance 
P'\  denote  the  tensions  of  the  intermediate  cords 
MN,  M'N',  by  P' ,  and  P" ;  the  radii  of  the  several 
wheels  by  p,  p  and  p" ;  and  those  of  the  correspond- 
ing axles  by  r,  r',  r" ;  then  we  shall  have 


STATICS.  89 


r      :    p, 
r'      :     p' ^ 

r"     :     p" ; 


P     :    P 

P     :    P' 
p.'/    .    pill 

and  hence 

P    :    P'    ::    rXr'xr"    :    pXp'xp"; 

that  is,  the  poioer  to  the  resistance,  as  the  product  of  the 
radii  of  the  axles  to  the  product  of  the  radii  of  the  wheels. 

66.  In  these  combinations,  the  connection  between 
the  simple  machines  is  frequently  effected  by  means 
of  teeth  or  cogs  projecting  from  the  several  convex 
surfaces,  as  represented  in  figure  51.  The  relation 
between  the  power  and  the  resistance  is  not  altered 
by  this  modification  of  the  mode  of  connection. 

THE    SCREW. 

67.  This  machine  consists,  1st,  of  the  interior  screw 

(usually  called  simply  the  screw),  a  cylindrical  solid, 

around  whose  convex  surface  passes  a  uniform  band 

or  fillet,  called  the  thread  of  the  screw,  oblique  to  the 

axis,  and  constantly  inclined  to  it  at  the  same  angle  ; 

and,  2d,  of  the  exterior  screw  or  nut,  a  hollow  cylinder 

of  the  same  diameter  as  the  solid  one,  on  the  concave 

surface  of  which  is  a  groove  exactly  adapted  to  the 

fillet  of  the  interior  screw.  The  mode  of  oj)eration 

of  this  machine  is  indicated  in  figures  52  and  53,  in 

which  AB  represents  the  interior  screw,  inserted  in 

the  nut  CD ;  and  OS,  O'S',  two  arms  or  levers  to 

which  the  forces  are  respectively  applied,  according 

as  it  is  desired  to  communicate  motion  to  the  nut  or 

to  the  interior  screw. 

12 


90  MECHANICS. 

The  investigation  of  the  properties  of  the  screw  is 
best  conducted  by  first  considering  the  manner  in 
which  the  fillet  may  be  conceived  to  be  generated. 

Let  ABDC  [Fig.  54]  be  a  cylinder,  and  BM  a  rect- 
angle, of  which  the  base  DM  is  equal  to  the  circum- 
ference of  the  base  of  the  cylinder.  Let  the  sides 
BD  and  NM  be  divided  into  the  equal  parts  BF,  FH, 
etc.,  NG,  GI,  etc. ;  and  let  the  oblique  lines  FN,  HG, 
etc.  be  drawn.  Then  if  the  rectangle  be  applied  to  the 
convex  surface  of  the  cylinder  in  such  a  manner 
that  the  line  NM  shall  coincide  with  BD,  the  points 
N,  G,  I,  etc.  will  coincide  with  the  points  B,  F,  H,  etc., 
and  the  oblique  lines  FN,  HG,  etc.  will  form  on  the 
surface  of  the  cylinder  a  continuous  curve  ss.  This 
curve  is  called  a  helix;  and  the  constant  interval 
NG,  its  pitch.  If  now  we  conceive  a  triangle,  whose 
plane  constantly  passes  through  the  axis  of  the 
cylinder,  to  revolve  about  that  axis  in  such  a  manner 
that  its  base,  in  no  case  greater  than  NG,  shall  con- 
stantly be  in  contact  with  the  cylinder,  and  have  one 
of  its  extremeties  in  the  helix  ss,  it  is  evident  that 
each  point  of  the  revolving  figure  will  describe  a  helix 
similar  to  ss\  and  that  the  assemblage  of  helices  thus 
described  will  form  the  fillet  of  a  screw.  The  gene- 
rating surface  is  usually  a  triangle,  as  we  have  above 
supposed  it  to  be  :  it  is  sometimes  a  rectangle,  and 
may  be  of  any  form  whatever.  The  conditions  of 
the  problem  being  the  same,  whether  we  suppose 
the  screw  to  be  movable  and  the  nut  fixed,  or  the 
converse,  we  will  adopt  the  latter  hypothesis. 


STATICS.  91 

Let  then  the  screw  be  supposed  to  be  fixed,  and 
the  nut  resting  upon  it  to  be  subjected  to  the  action 
of  two  forces  :  the  one,  the  resistance  R,  acting 
directly  upon  it  in  the  direction  of  the  axis ;  the 
other,  the  power  P,  acting  at  S  at  right  angles  to  the 
axis,  and  also  to  the  lever  OS.  In  the  revolution  of 
the  nut,  each  of  its  points  in  contact  with  the  screw 
describes,  as  we  have  seen,  a  helix  and  may  obviously 
be  regarded  as  moving  on  an  inclined  plane  whose 
height  is  equal  to  the  pitch  of  the  screw,  and  whose 
base  is  equal  to  the  circumference  of  the  circle  hav- 
ing for  its  radius  the  distance  of  the  point  from  the 
axis  of  the  screw.  Hence,  considering  a  single  point 
of  contact  M  [Fig.  55],  whose  distance  from  the  axis 
is  OM,  and  denoting  the  forces  which  act  upon  it  by 
p  and  r,  both  being  supposed  to  act  directly  upon 
the  point,  the  first  parallel  to  the  base  of  the  plane, 
the  second  perpendicular  to  it,  we  shall  have,  in  the 
case  of  an  equilibrium  [Art.  59,  2°], 

P    :    r    ::    NO    :    27rx  OM. 
But  if  the  power  be  applied  at  S  instead  of  M,  at  the 
distance  OS  from  the  axis,  we  shall  have,  to  deter- 
mine the  force  P  which,  applied  at  S,  will  be  equi- 
valent to  p  applied  at  M,  the  proportion 

P    :    i?    ::    OM    :    OS; 
whence  we  get 

P    :    p    ::    2:7 xOM    :    SttxOS. 

Comparing  this  proportion  with  the  first,  we  have 

P    :    r    ::    NG    :    27rx  OS; 
and  hence 


92  MECHANICS. 

«  NG 


r  X 


2-xOS 

For  the  other  points  of  contact  M',  M",  etc.,  at  whi  h 
the  resistances  r',  r",  etc.  are  put  in  equilibrium  by 
the  powers  P',  P",  etc.,  applied  at  the  same  distance 
from  the  axis  as  the  power  P,  we  have 

NG 


P'    ^  r'  X 
P'  —  r"  X 


2-xOS' 

NG 


SttXCS' 

etc. 
Hence,  denoting  the  sum  of  the  powers  P,  P',  etc. 
by  P,,  and  the  sum  of  the  resistances  r,  r',  etc.  by 
R,  we  have 

^      —    ^^^    2TrX0S   ' 

whence, 

P,    :    1?    : :    NG    :    2  TT  X  OS. 

That  is,  in  the  equilibrium  of  the  screw,  the  power  is 
to  the  resistance,  as  the  pitch  of  the  screw,  or  the  distance 
between  the  threads,  is  to  the  circumference  of  the  circle 
described  by  the  point  of  application  of  the  power. 

THE    WEDGE. 

68.  The  wedge  is  a  solid  body  of  the  shape  of  a 
triangular  prism.  The  surface  CDEF  [Fig.  56],  is 
called  its  back  ;  ABED  and  ABFC,  its  sides ;  and  AB 
the  intersection  of  the  sides,  its  edge.  The  use  to 
which  it  is  most  commonly  applied,  is  to  separate 
the  parts  of  a  body,  by  introducing  the  edge  AB  into 
a  small  cleft,  and  applying  an  impulsive  force  to  the 
back.  Since  the  resistance,  or  the  force  which  the 
parts  of  the  body  oppose  to  the  separation,  is  always 


STATICS.  93 

unknown,  it  would  be  useless  to  investigate  the  con- 
ditions of  equilibrium  between  it  and  the  power. 
We  shall,  therefore,  seek  merely  the  relation  between 
the  power  and  its  components  perpendicular  to  the 
sides  of  the  wedge. 

Let  MNO  [Fig.  57]  be  a  section  of  the  wedge  by 
a  plane  perpendicular  to  its  edge.  Let  the  power 
P,  which  we  suppose  to  be  perpendicular  to  NO,  be 
represented  by  IK,  and  be  resolved  into  the  two 
components  IR  and  IS  perpendicular  respectively 
to  NM  and  OM.  These  components  represent  the 
effects  of  the  power  at  the  sides  of  the  wedge,  and 
tend  directly  to  separate  the  parts  of  the  body. 
Denoting  the  power  and  its  components  by  P,  P' 
and  P"  respectively,  we  have 

P    :    P'    :    P"    ::    IK    :    IR    :    IS. 

But  from  the  similar  triangles  MNO  and  IKE,  we 

have 

IK    :    IR    :    IS    : :    NO    :    MN    :    MO: 
hence 

P    :    P'    :    P"   ::    NO    I    MN    :    MO; 

and  multiplying  the  last  three  terms  of  this  propor- 
tion by  the  line  DE  [Fig.  56],  we  get 
P    '.     P'     :     P"     ::     NO  X  DE     :     MN  x  DE     :     MO  X  i'E. 

The  products  composing  the  last  three  terms  of 
this  proportion  represent  the  respective  surfaces  of 
the  back  and  sides  of  the  wedge.  Consequently,  in 
the  wedge,  the  power  applied  at  right  angles  to  the  back, 
and  the  efforts  exerted  at  the  sides,  are  respectively  pro- 
portional to  the  surfaces  of  the  hack  and  sides. 


94  MECHANICS. 

GENERAL    PRINCIPLE    OF    EQUILIBRIUM    IN    MACHINES. 

69.  Bj  combining  the  simple  machines  above 
described,  an  endless  variety  of  compound  machines 
may  be  formed.  In  these  compound  machines,  the 
ratio  of  the  power  to  the  resistance  can  be  deter- 
mined when  the  tensions  of  the  cords  which  connect 
the  various  parts  are  known,  as  has  been  done  in  the 
case  of  the  system  of  pulleys  of  article  63,  and  that 
of  wheels  and  axles  of  article  65.  But  in  every  case, 
however  complex  the  machine,  this  ratio  can  also  be 
determined  by  the  following  rule: 

liCt  the  equilibrium  of  the  machine  be  supposed 
to  suffer  an  infinitely  small  disturbance  :  the  points 
of  application  of  P  and  R,  the  power  and  the  resist- 
ance, will  describe  infinitely  small  arcs.  If  the  direc- 
tions of  P  and  R  are  tangent  to  these  arcs,  let  the 
arcs  be  denoted  by  u  and  v  respectively ;  if  they  are 
not,  let  the  projections  of  the  arcs  upon  the  directions 
of  P  and  R  be  denoted  by  these  letters;  then  will 
the  relation  between  P  and  R  be  expressed  by  the 
proportion 

P     :     R     \:     V     :     It. 

This  rule  is  a  particular  case  of  a  general  principle 
of  mechanics,  called  the  principle  of  virtual  velocities. 

70.  Applications. 

1°  The  lever.  1st.  Let  AB  [Fig.  58]  represent  the 
lever ;  and  from  F  the  fulcrum,  let  the  perpendicu- 
lars p  and  p'  be  drawn  to  the  directions  of  the  two 
forces  P  and  P'.  Let  M  and  N,  the  feet  of  the  per- 
pendiculars, be  taken  as  the  points  of  application  of 


STATICS.  95 

P  and  P' ;  and  conceive  an  infinitely  small  motion 
to  be  communicated  to  the  lever,  causing  these 
points  to  describe  the  infinitely  small  arcs  Mw  and 
^n.  The  directions  of  P  and  P'  being  tangent  to 
these  arcs,  denoting  them  by  u  and  v,  we  shall  have, 
by  the  rule, 


P 

P'    : 

:      V 

:     u 

but 

V 

u      : 

:     p' 

:     p 

hence 

P 

P'     : 

:     p' 

•     P 

That  is,  the  forces  are  inversely  as  the  perpendicu- 
lars drawn  from  the  fulcrum  to  their  directions  as 
found  in  article  55. 

2d.  Let  the  forces  P  and  P  be  supposed  to  be 
applied  at  the  points  A  and  B  [Fig.  59]  the  extremi- 
ties of  the  lever.  In  the  case  of  motion,  the  direc- 
tions of  P  and  P  will  not  in  general  be  tangent  to 
the  arcs  AE  and  BG  described  by  A  and  B ;  and  in 
applying  the  rule,  we  must  consider  u  and  v  as 
denoting  the  projections  of  AE  and  BG  upon  these 
directions. 

To  find  the  values  of  u  and  i;,  draw  EC  and  GD 

perpendicular  to  the  directions  of  P  and  P ,  and 

consider  the  infinitely  small  arcs  AE  and   BG  as 

straight  lines  perpendicular  to  the  radii  FA  and  FB  ; 

then  we  have  the  triangles  MFA,  CAE  similar  to 

each  other,  and  also  the  triangles  NFB,  DBG ;  and 

hence  we  get 

FA    :    FM     :  :     AE    :    AC  or  u, 

and  FB    :    FN     :  :     BG    :    BD  or  i; ; 

A  T<^  T^P 

and  hence        u  =  — -  x  FM,     and  v  =  -— -  X  FN. 
i^  A  r  £) 


96  MECHANICS. 

Applying  the  rule,  we  have 

P     :    P-     ::    f|- X  FN    :    fl  X  FM 

but  AE    :    BG    :  :    FA    :    FB, 

BG  AE 


or 

consequently 


FB    ""  FA 


P    :     P'     :  :     FN    :     FM, 

or  P    :     P'     ::      p'      :      p. 

2"  Wheel  and  axle.  In  the  wheel  and  axle,  the 
directions  of  the  forces  P  and  R  [Fig.  49]  are  tangent 
to  the  arcs  described  by  their  points  of  application  : 
hence,  denoting  the  arcs  by  u  and  v  respectively,  we 
have,  by  the  rule, 

P    :    E    ::     V     :    u; 

but  V      :    u     :  '.     r     :    p : 

hence  P     :    R    :  :     r     \    p. 

3"  The  screw.  In  the  screw,  the  direction  ST 
[Fig.  52]  of  the  power  is  not  tangent  to  the  arc  of 
the  helix  described  by  S  its  point  of  application. 
Hence,  in  the  proportion 

we  must  regard  u  as  denoting  the  projection  upon 
ST  of  an  infinitely  small  arc  described  by  the  point 
S ;  V  being  the  corresponding  space  described  by 
any  point  of  the  nut,  in  the  direction  of  the  axis  of 
the  screw.  Conceive  a  plane  to  pass  through  OS 
perpendicular  to  the  axis,  and  let  the  helix  described 
by  the  point  S  be  projected  upon  this  plane ;  the 
projection  will  be  the  circumference  of  the  circle  of 
which  OS  is  the  radius.  Now  since  ST  is  tangent  to 


STATICS.  97 

the  circumference  of  the  circle,  the  projection  of  the 
infinitely  small  arc  of  the  helix  upon  this  line  will 
be  equal  to  its  projection  upon  the  circumference, 
and  the  one  may  be  taken  for  the  other.  Considering 
then  u  as  an  arc  of  the  circle,  it  is  evident  from  the 
nature  of  the  helix,  that  v  is  to  u,  as  the  pitch  NG 
[Fig.  55]  of  the  screw  is  to  the  circumference  of  the 
circle  of  which  OS  [Fig,  52]  is  the  radius,  or  that 

V     :     w     ::     NG    :     2  7rX  OS; 

whence  we  get 

P     :    7?    :  :     NG    :    2  TT  X  OS. 

4°  Systems  of  pulleys.  We  will  consider  the  system 
of  pulleys  represented  in  figure  48.  Denoting  the 
spaces  described  by  the  points  of  application  of  the 
forces  P  and  R  hy  u  and  v  respectively,  we  have, 
by  the  rule, 

P     :     R     ::     v    :    u. 

But  when  by  the  action  of  P,  the  weight  R  is  ele- 
vated through  the  space  m,  each  of  the  six  branches 
of  the  cord  embracing  the  movable  pulleys  is  short- 
ened by  the  same  quantity ;  and  hence  the  point  of 
application  of  the  power  must  descend  through  the 
space  6w.  Hence  we  have 


1     :     6, 


and  consequently 

P    :    R 


13 


98  MECHANICS. 

FRICTION.     ' 

71.  A  perfectly  smooth  body,  placed  on  an  inclined 
plane  also  perfectly  smooth,  would  move,  if  aban- 
doned to  the  action  of  gravity,  however  slight  the 
inclination  of  the  plane  ;  but  in  practice  this  motion 
never  takes  place,  till  the  angle  of  inclination  reaches 
a  certain  magnitude,  greater  or  less  according  to  the 
circumstances  of  the  case.  The  reason  of  this  is,  that 
the  surfaces  of  even  the  most  highly  polished  bodies 
are  only  comparatively  smooth,  and  consequently 
the  sliding  of  one  surface  upon  another,  is  always 
attended  with  a  certain  resistance.  This  resistance 
is  called  friction. 

Let  the  body  MN  [Fig.  60]  be  placed  on  a  plane 
AB,  and  let  the  angle  of  inclination  of  the  plane  be 
gradually  increased  till  it  reaches  the  magnitude  2, 
at  which  the  body  just  begins  to  move.  In  this 
position  of  the  plane,  let  P  the  weight  of  the  body 
applied  at  its  centre  of  gravity  C,  and  represented 
by  CF,  be  resolved  into  the  two  components 

CG-  =  /^cos  i,  and  CE  =  P  sin  z, 
the  one  perpendicular  to  the  plane,  the  other  paral- 
lel to  it.  The  first  will* obviously  measure  the  pres- 
sure of  the  body  upon  the  plane  ;  the  second,  its 
friction.  If  now  while  the  surface  in  contact  with 
the  plane  remains  the  same,  the  weight  P,  and 
hence  the  pressure  P  cos  z,  be  made  to  vary,  the 
angle  of  inclination  at  which  the  body  begins  to 
move  will  be  found  to  be  unaffected,  retaining 
constantly  its  first  value  ?  :  whence  we  infer, 


STATICS.  99 

1"  That  the  friction  is  directly  proportional  to  the 
'pressure. 

Denoting  the  friction  by  F,  the  pressure  by  F, 
and  the  ratio  of  the  two  by  /,  we  have 

J   —  jii  ~  *^^g^ 
and  F  =  F'  X  tang  i  =  F'  x  f. 

The  angle  i  is  called  the  angle  of  friction,  and  the 
constant/^/ie  coefficient  of  friction.  The  latter  expresses 
the  friction  for  the  unit  of  pressure,  and  is  taken  as 
the  measure  of  friction. 

If  the  body  MN  be  a  polyhedron  whose  faces  are 
unequal  in  extent,  but  equally  polished,  the  angle  i 
at  which  it  will  begin  to  move  will  be  found  to  be 
the  same,  on  whichever  face  it  may  rest  upon  the 
plane  :  whence  it  appears, 

2"  That  the  friction  is  independent  of  the  extent  of 
the  surface  in  contact. 

It  is  also  found, 

S""  That  the  friction  is  independent  of  the  velocity;  that 
is,  that  the  friction  is  the  same,  whatever  the  velo- 
city with  which  the  one  surface  moves  over  the 
other. 


PART  SECOND. 


DYNAMICS 


OF   THE  RECTILINEAR   MOTION  OF  A  MATERIAL   POINT. 

I.  Of  uniform  rectilinear  motion. 

1.  Uniform  rectilinear  motion  is  that  in  which  a 
material  point,  moving  in  a  right  line,  passes  over 
equal  spaces  in  equal  times.  It  is  the  simplest  kind 
of  motion,  and  takes  place  whenever  a  body  is  acted 
upon  by  an  impulsive  force,  and  then  abandoned  to 
itself. 

The  velocity  of  the  point  is  the  space  which  it  describes 
in  the  interval  of  time  arbitrarily  chosen  as  the  unit.  It 
evidently  expresses  the  rate  at  which  the  point 
moves. 

If  the  velocity  be  denoted  by  v,  then 
The  space  described  at  the  end  of  the  1st  unit  of  time,  is  r, 

u  a  u     2d        "  "  2v, 

"  "  ♦'    3d      "  "         3u, 


"  "  "    tth.     "  "         tv; 

and  denoting  the  whole  space  by  s,  we  have 

s  =  vt [1] 


102  MECHANICS. 

§  1.  From  the  preceding  equation,  we  get 


from  which  it  appears,  that  in  uniform  motion,  the 
velocity  is  the  ratio  of  the  space  described  to  the  time 
employed  in  describing  it. 

§  2.  If  we  denote  by  s  and  s"  the  spaces  passed 
over  in  the  times  t'  and  t"  by  two  points  moving 
uniformly  with  the  velocities  v  and  v",  we  have 

S'     =z    V't', 

s"  —  v"t"; 
and  hence  s'     :     s"     : :     v't'     :  v"t" ; 

that  is,  the  spaces  described  are  as  the  products  of  the 
times  by  the  velocities. 

When  the  times  are  equal,  we  have 

s'    :    s"    : :    v'    :    v", 

or  the  spaces  as  the  velocities. 

When  the  velocities  are  equal,  we  have 
s'    :    s"    ::    t'    :    t", 
or  the  spaces  as  the  times. 

When  the  spaces  are  equal,  we  have 
t'    :    t"    : :    v"    :    v', 
or  the  times  inversely  as  the  velocities. 

2.  A  more  general  form  may  be  given  to  equation 
[1].  Thus  [Fig.  1]  suppose  the  material  point  to  be 
moving  on  the  line  AB  from  left  to  right,  wdth  a 
velocity  v,  and  denote  by  s  its  distance  at  any  instant 
from  the  point  0  arbitrarily  assumed  on  AB,  and  let 
the  time  t  be  reckoned  from  the  instant  at  which  the 


DYNAMICS.  103 

point  is  at  0' ;  then  when  the  point  is  at  D  to  the 
right  of  0',  we  have 

OD  =  00'  +  O'D, 
or,  putting,  00'  =  6, 

s  =  vt-\-h [-2] 

In  this  equation,  the  variables  s  and  t  may  be  either 
positive  or  negative.  The  positive  vahies  of  t  refer 
to  times  posterior  to  the  instant  at  which  the  point 
is  at  0' ;  the  negative  values,  to  times  anterior  to 
the  same  instant.  The  positive  values  of  s  must  be 
reckoned  from  0  to  the  right ;  the  negative  values, 
from  0  to  the  left. 

In  this  general  form,  the  equation  will  enable  us 
to  determine,  at  any  instant  whatever,  the  position 
of  the  material  point  on  the  indefinite  straight  line 
AB. 

If  we  suppose  another  material  point  to  move  in 
the  line  AB  with  a  velocity  v\  and  to  be  at  0"  at  the 
instant  at  which  the  first  point  is  at  0',  its  motion 
will  be  determined  by  the  equation 

s'  z=z  v't-\-b' ;    [8] 

s'  denoting  its  variable  distance  from  O,  and  b'  the 
distance  00". 

By  means  of  equations  [2]  and  [3],  any  problem 
may  be  solved,  which  depends  upon  the  relative 
motions  of  the  two  points.  Thus,  to  determine  when 
the  two  points  will  meet  each  other,  in  which  case 

s  =  s', 
we  have  vt-\-b  z=  v't-\-b'j 

and  hence  t  ^= -. 


104  MECHANICS. 

II.  Of  motion  uniformly  varied. 

3.  A  force  which  acts  without  intermission,  and  with  a 
constant  intensity,  is  called  a  constant  accelerating  force. 
The  motion  of  a  material  point,  subjected  to  the 
action  of  a  constant  accelerating  force,  is  called  a 
uniformly  varied  motion. 

Let  a  material  point  be  supposed  to  be  acted  upon 
by  a  constant  accelerating  force  :  then  if  at  any 
instant  the  accelerating  force  cease  to  act,  the  motion 
will  evidently  become  uniform,  and  the  point  will 
move  with  the  velocity  which  it  had  at  that  instant. 
Hence  in  motion  uniformly  varied,  the  velocity  at 
any  instant  is  the  space  which  the  point  would 
describe  in  any  succeeding  unit  of  time,  should  the 
accelerating  force  at  that  instant  cease  to  act.  The 
velocity  evidently  depends  upon  the  intensity  of  the 
force,  and  the  time  during  which  it  has  been  acting. 

Since  a  constant  accelerating  force  acts  at  every 
instant  with  the  same  intensity,  it  must  generate 
equal  velocities  in  equal  times.  We  assume  as  the 
measure  of  a  constant  accelerating  force,  the  velocity  which 
it  generates  in  the  unit  of  time. 

Let  us  now  suppose  a  material  point  to  move  from 
a  state  of  rest,  under  the  action  of  a  constant  acce- 
lerating force,  of  which  the  intensity  is  denoted  by 
g ;  and  let  us  seek  the  velocity  v,  and  s  the  space 
described,  at  the  end  of  the  time  t^. 

To  facilitate  the  investigation,  conceive  the  time 

"^The  time  t  is  given  in  terms  of  the  unit  of  time;  that  is,  if  the 
second  be  taken  as  the  unit,  t  denotes  a  certain  number  of  seconds. 


DYNAMICS.  105 

divided  into  an  infinite  number  of  infinitely  small 
equal  intervals,  or  instants ;  and  suppose  the  force  to 
act  upon  the  point  at  the  commencement  of  each 
instant,  communicating  to  it,  at  each  repetition  of 
its  action,  an  infinitely  small  velocity.  Then  during 
each  instant  the  motion  will  be  uniform,  and  the 
variable  motion  will  be  resolved  into  a  series  of 
uniform  motions. 

Denote  by  n  the  number  of  instants  in  a  single 
unit  of  time,  and  by  k  the  number  of  instants  in  the 
whole  period  t;  then 

k  =  nt. 

Also  let  i  equal  the  space  which  the  point  describes 

during  an  instant,  in  virtue  of  the  constant  velocity 

which  the  force  communicates  to  it  at  the  beginning 

of  each  instant ;  then  the  spaces  described  by  the 

point  during  the  successive  instants  of  the  time  t, 

will  be 

t,     2i,     Si,  . . , ,  ni,  . . . ,  ki [4] 

If  at  the  end  of  the  first  second  the  accelerating 
force  should  cease  to  act,  the  point,  having  during 
the  nth.  instant  described  the  space  ni,  would,  during 
the  following  second,  describe  the  space  n.?ii  —  nri ; 
and  this  being  the  velocity  communicated  during 
the  preceding  second,  or  the  measure  of  the  accele- 
rating force,  we  shall  have 

g  z=  n'i. 

If  at  the  end  of  the  time  t  we  suppose  the  accele- 
rating force  to  cease  acting,  the  point,  which  in  the 

^th  instant  has  described  the  space  ki,  will,  in  the 

14 


106  MECHANICS. 

folloAving  second,  describe  the  space  nxki  =  nki ; 
and  this  will  be  the  velocity  acquired  at  the  end  of 
the  time  t.  Denoting  this  by  v,  we  shall  have 

V  =  nki; 
but  A:  =  7ct ; 

hence  v  i=  Tt^it 

=  gt [5] 

To  find  the  space  s,  we  have  only  to  determine 
the  sum  of  the  series  [4]  :  we  thus  have 

s  =  {i-{.hi)  — 

=  (i  +  ^)|; 
but  k  being  infinite,  (1  +  k)  becomes  k,  and  we  have 
s  =  ^  =  ^  =  ige [6] 

4.  We  have  supposed  the  material  point  to  be  at 
rest  when  the  accelerating  force  begins  to  act  upon 
it  :  if  we  suppose  it  to  be  in  motion,  and  to  have 
already  described  a  space  b  with  a  constant  velocity 
a,  we  shall  have  the  more  general  equations 

V  =  a  +  gt,  [7] 

s   =  b  +  at+yt^ [8] 

If  the  accelerating  force  act  in  the  direction 
opposite  to  the  primitive  impulse,  g  must  be  affected 
with  the  negative  sign.  The  motion  in  this  case  is 
said  to  be  uniformly  retarded. 

If  we  suppose  g  =  0,  the  equations  are  reduced 
to  those  of  uniform  motion. 

5.  If  a  material  point,  moving  from  a  state  of  rest 


DYNAMICS.  107 

under  the  action  of  a  constant  accelerating  force  g, 
describe  the  spaces  s  and  s'  in  the  times  t  and  t',  and 
acquire  the  velocities  v  and  v'  equations  [5]  and  [6] 
give 

V  —  gt,  v'  =  gt' ; 

s   =   i^gt\      s'  —   Igt"; 

and  hence  we  have  the  proportions 


s    :    s' 

V   :    v' 


t 


VJ'; 


that  is,  in  motion  uniformly  accelerated,  the  spaces  de- 
scribed are  as  the  squares  of  the  times,  and  the  velocities 
acquired  are  as  the  times,  or  as  the  square  roots  of  the  spaces, 

6.  If  in  the  equation  s  —  i  gf ,  we  make  ^  =  1, 
we  get 

s  =  l^,  or  ^  =  2s; 
that  is,  the  velocity  acquired  in  the  unit  of  time  is 
equal  to  twice  the  space  described  during  that  time ; 
and  hence,  as  the  unit  of  time  is  arbitrary,  it  follows 
that  a  constant  accelerating  force  communicates  to  a  ma- 
terial point,  in  any  time  whatever,  a  velocity  equal  to  twice 
the  space  which  it  causes  the  point  to  describe  in  the  same 
time. 

7.  It  can  be  shown  by  direct  experiment,  that 
bodies,  near  the  surface  of  the  earth,  fall  with  a 
uniformly  accelerated  motion.  Terrestrial  gravity 
may  therefore  be  considered  a  constant  accelerating 
force ;  and  hence  equations  [7]  and  [8]  may  be 
employed  to  determine  the  circumstances  of  the 
motion  of  falline:   bodies.    To  find   the   numerical 


108  MECHANICS. 

value  of  g  when  it  rej)resents  the  intensity  of 
gravity,  we  have  recourse  to  an  indirect  process, 
which  will  be  explained  hereafter.  It  has  thus  been 
found,  that  in  the  latitude  of  the  city  of  New-York, 
tlie  second  being  assumed  as  the  unit  of  time. 

g  —  32,1598  feet,  or  nearly  32^  feet. 

8.  If  from  the  equations 

s  =   \gf  and  V  =  gt, 

we  eliminate  t,  we  find 

V  —  s/2ii,  [9] 

an  equation  which  gives  the  velocity  acquired  in 
falling  through  a  given  height,  or,  as  it  is  usually 
expressed,  the  velocity  due  to  a  given  height. 

To  determine  the  time  in  which  a  body  will  fall 
through  a  given  height,  we  have  the  equation 

t^  v/- [^^^ 

V  g 

9.  To  determine  the  circumstances  of  the  motion 
of  a  body  projected  vertically  upwards  :  in  equations 
[7]  and  [8],  we  make  6  —  0,  and  affect  g  with  the 
negative  sign ;  we  thus  get 

»  =  a  —  gt [a] 

s   =  at—lgt\    [bj 

in  which  a  denotes  the  velocity  of  projection. 

To  find  the  time  during  which  the  body  is  ascend- 
ing :  in  equation  [a],  we  make  t;  =r  0 ;  we  thus  get 


To  find  the  greatest  elevation  of  the  body,  we 


DYNAMICS.  109 

substitute  this  value  of  t  in  equation  [b]  :  we  thus 
get 

The  body  having  attained  its  greatest  elevation, 
to  find  the  velocity  which  it  will  acquire  during  its 
descent  :  in  the  equation  v  =  \^Hs,    we  substitute 

for  s  the  expression  ^ ;  we  thus  get 

From  this  last  result,  it  appears  that  the  velocity  ac- 
quired during  the  fall  is  equal  to  the  velocity  of  projection. 

10.  A  force  acting  without  intermission,  and  constantly 
varying  in  intensity  during  the  time  of  its  action,  accord- 
ing to  some  law,  is  called  a  variable  accelerating  force. 

The  velocity  of  a  point  moving  under  the  action 
of  a  variable  accelerating  force,  is  measured  in  the 
same  manner  as  in  motion  uniformly  varied. 

The  measure  of  the  intensity  of  a  variable  accele- 
rating force,  at  any  instant,  is  the  velocity  which  it 
would  generate,  should  it  act  during  the  unit  of  time 
with  the  intensity  which  it  has  at  that  instant.  Thus 
a  point  being  in  motion  under  the  action  of  a  varia- 
ble accelerating  force,  to  determine  the  measure  of 
the  force  at  the  end  of  the  time  t,  conceive  the  force 
at  that  instant  to  cease  varying,  and  to  act  during 
the  succeeding  second  with  the  intensity  which  it 
has  at  that  instant ;  the  velocity  acquired  hy  the 
point  during  the  second  is  the  measure  of  the  force. 
Let  this  velocity  be  denoted  by  </> ;  then  the  velocity 
acquired  during  the  infinitely  small  interval  of  time 


110  MECHANICS. 


6,  immediately  succeeding  /,  is  evidently 'i/';  and 
denoting  this  velocity,  which  is  also  infinitely  small, 
by  v',  we  have  (pf  =  v\  and  hence 

<P  -T' 
For  the  case  of  a  constant  accelerating  force,  we 
have 

V 

^  =  T' 
Hence  the  measure  of  an  accelerating  force  is  the 
same,  whether  it  be  constant  or  variable ;  only  in 
the  latter  case  the  time  and  velocity,  of  which  the 
ratio  is  taken,  are  infinitely  small. 

Experiments  made  near  the  surface  of  the  earth 
indicate,  as  already  remarked,  no  variation  in  the 
intensity  of  gravity  at  different  points  of  the  same 
vertical.  A  more  extended  induction,  however,  shows 
that  gravity  is  really  a  variable  force.  The  general 
law  is,  as  will  be  seen  in  a  subsequent  article,  that 
the  intensity  of  gravity  depends  upon  the  distance 
of  the  point  at  which  it  acts,  from  the  centre  of  the 
earth ;  the  intensities  at  any  two  points  of  the  same 
vertical  being  inversely  as  the  squares  of  the  dis- 
tances. It  can  be  shown,  by  a  very  simple  calculation, 
that  according  to  this  law,  the  motion  of  a  falling 
body  near  the  surface  of  the  earth  should  not  differ 
sensibly  from  a  uniformly  accelerated  motion. 


DYNAMICS.  Ill 

OF    THE    MOTION    OF    BODIES    UPON    INCLINED    PLANES. 

11.  The  motion  of  a  body  placed  on  an  inclincxl 
plane,  and  abandoned  to  the  action  of  gravity,  may 
evidently  be  rednced  to  that  of  a  material  point  nnder 
the  same  circumstances."^ 

To  consider  the  motion  of  a  material  point  on  an 
inclined  plane,  conceive  the  inclined  plane  to  be  inter- 
sected at  right  angles  by  a  vertical  plane,  and  suppose 
the  section  to  be  represented  by  AB  [Fig.  2] ;  and 
from  A  and  B  let  the  lines  AC  and  BC  be  drawn,  ver- 
tical and  horizontal  respectively.  Denote  the  angle 
at  B  by  i,  and  the  height  and  length  of  the  plane  by 
h  and  /  respectively.  A  material  point,  placed  on  the 
plane  at  A,  will  evidently  describe  the  line  AB. 

Suppose  the  point  to  be  at  M ;  and  let  g,  the  inten- 
sity of  gravity,  be  represented  by  the  vertical  MP. 
Let  MP  be  resolved  into  the  two  components  MR 
and  MQ,  the  one  in  the  line  AB,  the  other  perpen- 
dicular to  it ;  the  latter  will  be  destroyed  by  the  reac- 
tion of  the  plane,  and  the  former  will  cause  the  j)oint 
to  describe  the  line  AB. 

The  angle  MPR  being  equal  to  the  angle  at  B,  we 
have 

MR  =  g  sin  i. 

*  It  may  be  proper  to  remark,  that  throughout  the  following  pages 
the  body  whose  motion  is  the  subject  of  investigation  is  considered  to 
be  under  the  influence  of  the  enumerated  forces  alone.  Thus,  in  the 
present  case,  it  is  supposed  to  be  acted  upon  by  gravity  only,  abstrac- 
tion being  made  of  friction  and  the  resistance  of  the  medium,  in  which 
the  motion  takes  place. 


112  MECHANICS. 

Thus  the  force  which  urges  the  material  point  dowi; 
the  incluied  plane  is  of  the  same  nature  as  gravity, 
differing  from  it  only  in  intensity.  If,  therefore,  in 
the  equations  of  uniformly  varied  motion, 

V  =  gt,     s  =  ^igt"^  and  c  —  \/-2gfi, 

we  substitute  g  sin  i  in  place  of  g,  we  shall  have,  to 
determine  the  circumstances  of  the  motion  of  the 
point,  the  equations 

v  =  gt  sin  z, [1  i j 

s  =  Igf  sin  2, [12] 

V  =  ^2gs  sin  i, [13] 

If  the  point  be  made  to  ascend  the  plane  by  an 
impulse  in  the  direction  BA,  the  motion  will  b. 
determined  by  the  equations 

v  =  a  —  gt  sin  2, 
s  =  at  —  ^gt"^  sin  /, 

a  denoting  the  velocity  due  to  the  impulse. 

12.  To  determine  the  velocity  acquired  by  the 
material  point  in  describmg  the  line  AB,  we  have  the 
equation 

V  =  \/2gs  sin  i 
=  \^2gl  sin  i 

But  \^2gh  is  the  velocity  which  the  point  would 
acquire  in  falling  through  the  height  h  or  AC  :  hence 
it  appears  that  a  material  point  acquires  the  same 
velocity  in  descending  the  length  of  an  inclined  plane, 
that   it  would  acquire  in  fiilling  freely  through  the 


DYNAMICS.  113 

height  of  the  plane ;  and,  consequently,  that  if  several 
material  points,  setting  out  from  the  same  point  A 
[Fig.  3],  and  moving  on  different  inclined  planes, 
describe  the  lines  AB,  AB',  AB",  on  arriving  at  the 
points  B,  B',  B",  situated  in  the  same  horizontal 
plane,  they  will  all  have  acquired  the  same  velocity. 
13.  To  find  the  relations  of  the  times  t  and  t'  em- 
ployed in  describing  the  lengths  /  and  /'  of  two  inclined 
planes  of  which  the  heights  are  h  and  A' ,  we  get  from 
equation  [12] 


f  = 


^g  .  sin  i 


and  hence 


I 


Hence,  for  the  second  plane,  we  have 
, 1 

and  consequently  we  have 

that  is,  the  times  employed  hy  two  material  points  in  des- 
cribing two  different  inclined  planes,  are  as  the  lengths 
of  the  planes  divided  by  the  square  roots  of  the  heights. 

14.  Let  AB  [Fig.  4J  represent  the  line  described 
by  a  material  point  in  descending  an  inclined  plane ; 
and  from  D  any  point  whatever  of  the  altitude  AC, 
draw  DE  perpendicular  to  AB.  Then  denoting  the 
angle  at  B  by  z,  and  by  ^  the  time  in  which  a  material 
point  will  describe  the  line  AE,  we  have  [equa.  12 

15 


114  MECHANICS. 

AE  =  i^i'  sin  i ; 
but  AE  =  AD  .  sin  i 


hence  tW^'  sin  i  =  AD  .  sin 


25' 


and  ^_  IAD 

Again,  denoting  by  V  the  time  in  which  a  material 
point  will  describe  the  line  E  D,  w^e  shall  find  by  a 
similar  operation, 

ITd 

J\D 
^Y-  is  the  exj)ression  of  the  time  [equa.  10]  in 

which  a  material  point,  moving  freely,  will  descend 
from  A  to  D ;  and  the  point  E  is  in  the  circumference 
of  the  circle  described  on  AD  as  a  diameter :  hence 

a  material  pointy  moving  in  the  plane  of  a  vertical  circle^ 
will  describe  any  chord  draion  from  either  extremity  of 
the  vertical  diameter,  in  the  same  time  that  it  will  des- 
cribe the  diarneter  itself. 


OF  THE   MOTION  OF  A  MATERIAL  POINT    ON  A  GIVEN   CURVE. 

15.  From  any  point  M  [Fig  5]  of  the  circumference 
of  a  circle  ABD,  let  the  straight  line  MP  be  drawn 
perpendicular  to  the  diameter  AB ;  then 

AP     :     PM     :  :     PM     :     PB. 
If  we  suppose  the  arc  BM  to  be  infinitely  small,  Ave 
shall  have  PM  infinitely  small  wdth  respect  to  AP, 


DYNAMICS.  115 

and  hence  PB  infinitely  small  with  respect  to  PM ; 
that  is,  PB  will  be  an  infinitely  small  quantity  of  the 
second  order.  But  PB  is  the  versed  sine  of  the  arc 
BM ;  hence  the  versed  sine  of  an  infinitely  small  arc 
is  an  infinitely  small  quantity  of  the  second  order. 

16.  Let  AB,  BC  [Fig.  6J  be  two  contiguous  sides 
of  the  perimeter  of  a  polygon ;  and  let  a  material 
point,  which  is  constrained  to  describe  this  perimeter, 
be  supposed  to  have  arrived  at  B  with  a  velocity  v. 
Let  V  be  represented  by  BD  a  portion  of  AB  produced 
and  be  resolved  into  the  two  velocities  BN,  BM,.  the 
one  in  the  direction  of  BC,  the  other  perpendicular 
to  it :  the  latter  is  destroyed  by  the  reaction  of  the 
side  BC ;  the  former  is  the  velocity  with  which  the 
point  describes  this  side. 

The  velocity  lost  by  the  point  in  passing  from  the 
side  AB  to  BC  is  evidently  equal  to  BD  —  BN.  But 
denoting  the  angle  NBD  by  u,  we  have 

BN  =  V  cos  u ; 

hence  we  have 

the  velocity  lost  =v  —  vcosu 

=  V  {1  COS  u) 

=  V  .  vers  sin  u. 

If  we  now  suppose  the  given  polygon  to  become 
a  polygon  of  an  infinite  number  of  infinitely  small 
sides,  or  a  curve,  the  angle  u,  which  any  side  makes 
with  the  adjacent  side  produced,  will  become  infi- 
nitely small;  and  hence  (Art.  15)  the  velocity  lost 
in  passing  from  the  one  side  to  the  other  will  be  an 


116  MECHANICS. 

infinitely  small  quantity  of  the  second  order.  When, 
therefore  a  material  point  is  constrained  to  describe 
a  curve,  under  the  action  of  either  impulsive  or  accel- 
erating forces,  it  retains  all  the  velocity  which  is  com- 
municated to  it  in  a  direction  tangent  to  the  curve ; 
for  since  the  velocity  lost  at  each  angle  of  the  polygon 
is  an  infinitely  small  quantity  of  the  second  order,  the 
whole  velocity  lost  in  passing  over  a  curve  of  finite 
length  can  only  be  the  sum  of  an  infinite  number  of 
infinitely  small  quantities  of  the  second  order,  that  is, 
an  infinitely  small  quantity  of  the  first  order. 

17.  Let  ABCD,  etc.  [Fig.  7]  be  a  vertical  curve, 
or  polygon  of  an  infinite  number  of  infinitely  small 
sides  AB,  BC,  CD,  etc. ;  and  let  the  sides  be  produced 
to  meet  the  horizontal  line  KH  in  A,  E,  H,  etc.  Then 
a  material  point  starting  from  A,  and  constrained  to 
describe  the  perimeter  ABCD,  etc.,  under  the  action 
of  gravity,  will  have,  when  it  arrives  at  the  point  B, 
the  same  velocity  as  if  it  had  described  the  line  EB 
[Art.  12] ;  and  as  by  the  preceding  article  it  loses  no 
velocity  in  passing  from  AB  to  BC,  when  it  arrives 
at  C,  it  will  have  the  same  velocity  as  if  it  had 
described  the  line  EC.  In  the  same  way  it  may  be 
shown  that  at  D  it  will  have  the  same  velocity  as  if 
it  had  described  the  line  HD,  or  the  vertical  GD. 
Hence,  generally,  a  material  point  which  descends 
under  the  action  of  gravity,  along  a  vertical  curve, 
has,  at  any  point  Avhatever,  the  same  velocity  as  if  it 
had  fallen  freely  from  a  height  equal  to  that  of  the 
arc  described.     It  is  also  evident  that  this  velocity  is 


DYNAMICS.  117 

independent  of  the  form  of  the  curve  ;  so  that  several 
material  points,  setting  out  from  the  same  point  0 
[Fig  8],  and  describing  in  the  descent  different  curves 
OM,  OM'  OM",  etc.,  will,  on  arriving  at  the  same 
horizontal  plane  MM' ' ,  have  the  same  velocity. 

18.  Let  AM  A'  [Fig.  9]  be  a  curve  symmetrical 
with  respect  to  the  vertical  MG,  in  which  the  points 
A  and  A'  are  in  the  same  horizontal  line,  and  TT'  is 
the  horizontal  tangent ;  and  su23pose  a  material  point, 
placed  on  the  curve  at  A,  to  be  abandoned  to  the 
action  of  gravity.  It  will  acquire,  in  descending  to 
the  point  M,  a  velocity  due  to  the  height  GM,  and, 
in  virtue  of  this  velocity,  will  rise  in  the  branch  MA' . 
But  the  velocity  with  which  it  will  describe  the  arc 
MA'  will  be  constantly  diminished  by  the  action  of 
gravity,  and  evidently  in  such  a  manner  that,  at  any 
point,  as  m! ,  its  velocity  will  be  the  same  that  it  was 
at  m,  m  and  w'  being  in  the  same  horizontal  line : 
hence  at  A'  its  velocity  will  be  reduced  to  zero.  From 
A'  the  point  will  descend  to  M,  and  acquire  as  before 
the  velocity  due  to  the  height  GM ;  and  this  velocity 
will  just  serve  to  carry  it  back  to  the  point  A  from 
which  it  set  out,  and  thus  it  will  continue  to  oscillate 
for  an  indefinite  time  between  A  and  A' .  Since  any 
element  of  the  curve  is  described  with  the  same  velo- 
city, whether  the  point  is  ascending  or  descending,  1l 
is  clear  that  the  time  of  the  ascent  in  either  of  the 
branches  is  equal  to  the  time  of  the  descent  in  the 
same  branch  :  it  is  also  evident  that  the  tAvo  branches 
are  described  in  equal  times.     The  point  will  there- 


118  MECHANICS. 

fore  occupy  the  same  time  in  going  from  A  to  A' ,  as 
in  returning  from  A'  to  A ;  and  thus  all  the  oscilla- 
tions will  be  made  in  equal  times.  Oscillations  Avhich 
are  thus  made  in  equal  times,  are  said  to  be  isochro- 
nous. 


OF    THE    SIMPLE    PENDULUM. 

19.  A  pendulum  is  an  apparatus  consisting  of  a  solid 
body  attached  to  one  extremity  of  a  rod,  through  the 
other  extremity  of  which  passes  a  horizontal  axis, 
about  which  the  whole  system  is  capable  of  oscillating. 

When  the  vertical,  drawn  through  the  centre  of 
gravity  of  the  system,  meets  the  axis  of  rotation,  the 
pendulum  is  at  rest ;  but  if  it  be  removed  from  this 
position,  and  abandoned  to  the  action  of  gravity,  it 
makes  a  series  of  oscillations,  which  may  be  show^n  to 
be  isochronous,  and  consequently  may  be  employed 
to  measure  time.  Such  a  pendulum  is  called  a  com- 
pomid  pendulum.  To  acquire  a  knowledge  of  its  prop- 
erties, we  first  investigate  those  of  a  purely  ideal 
pendulum,  called  the  simple  pendulum.  In  this  pen- 
dulum, the  solid  body  of  the  compound  pendulum  is 
supposed  to  be  reduced  to  a  heavy  material  point, 
and  the  rod  to  a  line  inextensible  and  inflexible,  and 
without  gravity. 

Let  CB  [Fig.  10]  be  a  simple  pendulum,  suspended 
at  C  ;  and  let  it  be  withdrawn  from  the  vertical  posi- 
tion CK,  and  made  to  assume  the  position  CB.     Let 


DYNAMICS.  119 

the  intensity  of  gravity,  represented  by  BN,  be  re- 
solved into  the  two  components  BQ  and  BP,  the  one 
in  the  direction  of  CB,  the  other  perpendicular  to  it. 
Of  these  two  components,  the  latter  alone  communi- 
cates motion  to  the  pendulum,  the  former  expending 
itself  in  producing  a  pressure  upon  the  point  C.  The 
case  is  then  precisely  similar  to  that  of  a  heavy  mate- 
rial point  compelled  to  describe  a  curve,  the  reaction 
of  the  fixed  point  taking  the  place  of  the  resistance 
of  the  curve. 

Suppose  the  pendulum,  or  material  point,  to  have 
already  descended  from  B  to  M  [Fig.  11],  and  to  have 
acquired  in  the  descent  a  velocity  u.  Draw  the  hori- 
zontal line  BD,  and  the  vertical  diameter  HK.  Also 
from  the  extremities  of  the  arc  MM' ,  supposed  to  be 
infinitely  small,  draw  MP,  M'  P'  perpendicular  to  HK ; 
and  on  AK  as  a  diameter,  describe  the  circumference 
ANO.  Draw  also  M'O'  perpendicular  to  MP,  and 
join  C  and  M' . 

Let  AP'  =  j:,  MT'  =  ?/,  PP' =  5,  AK  =  5, 
M'  C  =  / ;  and  denote  the  time  in  wdiicli  the  pen- 
dulum describes  the  arc  MM'  by  V . 

On  account  of  the  smallness  of  the  arc  MM' ,  w^e 
may  suppose  the  velocity  (u)  not  to  vary  while  the 
point  passes  from  M  to  M':  hence  we  immediately 
have 

u 

But  the  triangles  M'CP',  MM'O',  having  their  sides 
mutually  perpendicular,  are  similar,  and  give 


120 

MECHANICS. 

MT 

'     :     M'C     :  :     M'O'     : 

MM'; 

and  hence 

^^^^              M'P' 

PP'  X  M'C 

-        MT'              '  • 

I 

y  ' 

and  [Art.  17] 

u  =  V'2gx, 

whence 

y        V2ga: 

Now 

(M'P'r  =  P'KxP'H, 

or 

t,  =  s/  (b  —  x)X  a 

2l  —  (b  —  x)): 

[e] 


and  if  we  suppose  the  oscillations  of  the  pendulum  to 
be  made  in  very  small  arcs,  (b  —  x)  may  be  neglected 
in  comparison  with  2/,  and  we  shall  have 

y  =  V  (b  —  x)2l. 

Substituting  this  value  of  y  in  the  expression  for  f , 
we  get 


\/2gx  .  s/2i  (6  —  x) 
f^g  .  ^x  (6  —  re) 


X 


Wg       s/xQ}  —  x) 

But,  applying  the  result  contained  in  equation  [c]  to 
the  circle  ANKO,  we  get 

V  v/  _  PP'  X  jAK  _    PP^  X  |AK \hs^ 

^  ^    -       :^^/p/        ~x/AP  X  P'K~/v/x"(6^^)' 

and  hence  we  have 

,,      v/Z       NN' 

Now  for  the  time  of  describing  each  of  the  indefi- 
nitely small  arcs  composing  the  finite  arc  BMK,  a 


DYNAMICS.  121 

similar  expression  may  be  found;  thus,  for  f '  the 
time  of  describing  the  indefinitely  small  arc  M'  M" , 
we  should  get 

Hence  if  we  denote  the  time  of  describing  the  whole 
arc  BKD  by  T,  we  shall  have 


IT- 

'^8 

X- 

ANK 
b        ' 

T-- 

2  arc  ANK 

=  TT     . 

VI 

and  _  

[14] 

TT  being  the  ratio  of  the  circumference  of  a  circle  to 
its  diameter. 

From  this  expression,  it  appears  that  when  a  pen- 
dulum oscillates  in  very  small  arcs,  the  time  of  an 
oscillation  is  independent  of  the  height  AK  which 
determines  the  extent  of  the  arc  described ;  and  hence 
that  oscillations  of  small  extent  will  be  isochronous, 
though  the  amplitudes  should  vary. 

20.  Let  /  and  /'  be  the  lengths  of  two  pendulums 
which  oscillate  in  the  times  T  and  T",  under  the 
action  of  gravity  of  different  intensities  denoted  res- 
pectively by  g  and  g';  then  we  shall  have 

^^^Jy.   and  T'^n^^. 

and  hence      r    :     T'     :  :     Jy     :    J|-;    [d] 

that  is,  the  times  of  oscillation  are  directly  as  the  square 

16 


122  MECHANICS. 

roots  of  the  lengths  of  the  pendulums,  and  inversely  as 
the  square  roots  of  the  intensities  of  gravity. 

§  1.  If  the  intensities  of  gravity  are  the  same,  we 
have 

and  if  n  and  n^  denote  the  number  of  oscillations  made 
by  the  two  pendulums  in  a  given  time  k,  then 

h  k 

T=  — ,    and  T'  =  ~, 

n  n 


and  A    :    _^ 

n  n' 


•     s/l     :     s/V 
or  n'"^     \     n-     '.  \     I     '.     l\ 


and  1  =  ^1^,1'. 

By  means  of  this  equation,  we  can  calculate  for  any 
place  the  length  of  the  pendulum  which  will  make  a 
given  number  of  oscillations  in  a  determmate  time, 
when  we  know  the  number  of  oscillations  which  a 
pendulum  of  given  length  makes  at  the  same  place  in 
the  same  time.  Any  small  error  that  may  be  made 
in  determining  the  time  k,  may  be  rendered  insensible 
by  taking  n  the  number  of  oscillations  sufficiently 
large.  It  is  thus  that  the  length  of  the  seconds  pen- 
dulum in  the  latitude  of  the  city  of  Ncav  York,  that 
is,  the  pendulum  which  in  that  latitude  makes  86400 
oscillations  in  a  mean  solar  day,  in  vacuo,  has  been 

found  to  be 

39,"'10168,  or  3,''-25847. 

§  2.  The  measure  of  the  intensity  of  gravity  (already 
given.  Art.  7),  may  now  be  found.  For  this  purpose, 
we  employ  the  equation 


DYNAMICS.  123 


From  it  we  get         p-—  !^-  ^. 

^  5  /TT2  ' 

and  making  t=  1",     7r=  3,1415926,  and  Z-  39,">  101G8, 

we  find  g  =  385,i'^-9183  =  32,ftl598. 

The  value  of  /,  and  consequently  that  of  g,  has  been 
found  to  vary  with  the  latitude  of  the  place. 

§  3.  If  we  suppose  the  lengths  of  the  two  pendu- 
lums to  be  equal,  proportion  [d]  becomes 

and  denoting  by  n  and  n'  the  number  of  oscillations 
made  by  the  pendulums  in  any  given  time,  we  have 

n'     :     n     :   :     s/ g'     :     s/g, 
or  g     '.     g'     '.   '.     n^     :     7i" ; [e] 

that  is,  the  intensities  of  gravity/  at  any  two  places  are 
to  each  other  as  the  squares  of  the  number  of  oscillations 
made  at  the  places  in  any  given  time,  either  by  the  same 
pendulum,  or  by  two  pendulums  of  equal  lengths. 

§  4.  The  two  pendulums  being  still  supposed  equal 
in  length,  if  we  denote  by  n  the  number  of  oscillations 
which  they  make  in  the  times  k  and  k ,  we  have 

h  Jc' 

T=   -,   and  T'  =  ~;     . 
n  n 

and  hence        ^o-    •    ^o-/    •  •    -^    •    A, 

i 
or  g     :     g'     :  :     k'     :     ¥ ..••[fj 

That  is,  the  intensities  of  gravity  at  any  two  places 
are  inversely  as  the  squares  of  the  times  employed  in 
making  the  same  number  of  oscillations. 


124  MECHANICS. 

From  either  of  the  proportions  [e]  and  [f  ],  it  ap- 
pears that  by  causing  the  same  pencluhim  to  oscillate 
at  places  in  different  latitudes,  the  relative  intensities 
of  gravity  at  these  places  can  be  obtained.  It  has  thus 
been  found  that  the  intensity  of  gravity  increases  as 
we  proceed  from  the  equator  towards  the  poles.  The 
relation  between  the  ellipticity  of  the  earth,  and  the 
intensity  of  gravity  at  different  points  of  its  surface, 
is  such  that  the  one  can  be  found  by  means  of  the 
other :  experiments  with  the  pendulum  thus  furnish  a 
means  of  determining  the  figure  of  the  earth. 

The  principle  expressed  in  these  proportions  is  also 
employed  to  determine  the  law  of  the  intensities  of 
the  electric  and  magnetic  forces. 


OF  CENTRAL  FORCES. 

21.  Let  a  material  point  be  supposed  to  be  in  mo- 
tion, in  virtue  of  a  primitive  impulse,  and  an  accele- 
rating force  which  constantly  solicits  it  towards  a  fixed 
point.  Let  the  accelerating  force  be  supposed  to  act 
at  infinitely  small  equal  intervals,  or  instants;  com- 
municating to  the  point,  at  the  commencement  of  each 
instant,  an  infinitely  small  velocity.  Let  C  [Fig.  12] 
Tje  the  fixed  point  through  which  the  direction  of  the 
accelerating  force  constantly  jDasses ;  and  su]3pose  the 
material  point  to  be  moving  in  the  line  MN,  with  a 
velocity  which  would  cause  it  to  describe  the  infinitely 
small  space   M'N   in  an   instant.     When  the  point 


DYNAMICS.  125 

arrives  at  M' ,  let  the  accelerating  force  be  supposed 
to  communicate  to  it  a  velocity  which,  were  it  at  rest, 
would  cause  it  to  describe  the  space  M'  G  in  an  instant , 
then  the  point  Avill  describe,  during  the  instant,  the 
diagonal  M'M"  constructed  on  M'N,  M'G.  Arrived 
at  M" ,  the  point,  if  left  to  itself,  would  move  in  M'  M' ' 
produced,  and  in  the  following  instant  describe  the 
line  M"N'  equal  to  M'M";  but  at  M",  the  accelerat- 
ing force  again  acts ;  and  if  we  represent  the  space 
which  it  alone  would  cause  the  point  to  describe  in  an 
mstant  by  M' '  G'' ,  we  shall  have  M"  M" '  for  the  space 
actually  described  during  the  instant.  It  may  thus 
be  shown,  that  as  long  as  the  accelerating  force  con- 
tinues to  act,  the  point  will  move  on  the  perimeter  of 
a  polygon,  the  sides  of  which,  being  infinitely  small 
and  all  in  the  same  plane,  form  a  plane  curve.  The 
plane  of  the  curve  is  evidently  that  which  contains  at 
the  same  time  the  fixed  point,  and  the  direction  of  the 
primitive  impulse. 

The  velocity  of  the  point,  at  any  instant,  is  the 
space  which  it  would  describe  in  the  direction  of  the 
element  of  the  curve  in  which  it  is  then  moving, 
should  the  accelerating  force  at  that  instant  cease  to 
act. 

The  line  drawn  from  the  fixed  point  C  to  the  posi- 
tion of  the  material  point  at  any  instant,  is  called  tlu 
radius  vector. 

22.  The  areas  described  by  the  radius  vector,  in 
any  two  consecutive  instants,  are  equal.  For  the  tri- 
angles MCM",  M"CN'  having   equal   ba^es  in  the 


126  MECHANICS. 

same  straiglit  line  and  a  common  vertex  C,  are  equal ; 
and  so  also  are  the  triangles  M"CN'  and  M"CM'", 
which  have  the  same  base  M"C,  and  their  vertices 
situated  in  the  same  straight  line  parallel  to  the  base. 
Hence  the  triangles  M'CM",  WCW",  that  is,  the 
spaces  described  by  the  radius  vector  in  two  consecu- 
tive instants,  are  equal.  But  the  areas  in  any  two 
consecutive  instants  being  equal,  the  areas  described 
in  any  two  equal  intervals  of  time  must  also  be  equal, 
since  each  interval  contains  the  same  number  of 
instants ;  and,  hence,  generally,  the  areas  described 
in  any  two  intervals  of  time  are  proportional  to  the 
intervals.  Thus  ivhen  a  material  point  describes  a 
curve,  in  virtue  of  a  primitive  impulse,  and  an  accelerat- 
ing force  which  constantly  solicits  it  towards  a  fixed 
point,  the  areas  described  by  the  radius  vector  are  pro- 
portional to  the  times  employed  in  describing  them. 

Conversely,  when  a  material  point,  which  describes 
about  a  fixed  point  a  plane  curve,  moves  so  that  the  radius 
vector  describes  areas  proportional  to  the  times,  the  direc- 
tion of  the  accelerating  force  constantly  passes  through 
the  fixed  point. 

For,  employing  the  preceding  figin^e,  let  C  be  the 
fixed  point  and  M'M",  M"M"'  the  infinitely  small 
spaces  described  in  two  consecutive  instants ;  then, 
since  tlie  areas  are  proportional  to  the  times,  we  have 
the  triangle 

CM"M'"  =  CM'M". 
But  if,  at  the   beginning  of  the  second  instant  the 
accelerating  force  should  not  act,  the  material  point 


DYNAMICS.  127 

would  describe  the  line  M"N'  in  M'M"  produced,  and 
equal  to  M'  M"  ;  this  force  then  must  [Statics,  Art.  9] 
be  equal  and  parallel  to  N'  M" ' .     But  the  triangle 

CM'M"=CM"N'; 
therefore,  CM"M'"  =  CM"N', 

and  hence  the  line  N'  M' "  which  joins  the  vertices  of 

these  triangles  must  be  parallel  to  the  common  base 

M"  C ;    consequently  the    accelerating  force   at  M' ' 

must  act  in  the  direction  of  M"  C. 

23.  Conceive  a  material  point  at  M  [Fig.  13],  to 

be  connected  with  a  fixed  point  C  by  a  thread  MC 

inextensible    and   without   mass,  and    suppose   it   to 

receive  an  impulse  in  the  direction  MT  at  right  angles 

to  MC.     In  virtue  of  this  impulse,  it  will  describe 

the  circumference  of  a  circle  of  which  the  fixed  point 

is  the   centre  and    MC  the  radius;    and  during  the 

motion,  the  thread  will  suffer  a  certain  tension  in  the 

direction  of  its  length.     But  if  we  suppose  applied  to 

the  point  an  accelerating  force  equal  to  this  tension, 

and  constantly  directed  towards  the  fixed  centre  C, 

we  may  consider  the  string  withdrawn,  and  the  point 

to  be  moving  freely  under  the  action  of  the  primitive 

impulse  and  the  accelerating  force.     Then  as  the  areas 

described  by  the  radius  vector  CM,  in  equal  times, 

will  be  equal  [Art.  21],  the  arcs  described  in  equal 

times  will  also  be  equal,  and  the  motion  of  the  point 

will  be  uniform ;  and  if  we  denote  the  velocity  of  the 

point  by  v,  and  by  s  the  space  described  during  the 

time  t,  we  shall  have 

s  =  vt. 


128  MECHANICS. 

Let  MN  be  the  arc  described  by  the  material  poin^ 
during  the  infinitely  small  interval  f ,  and  denote  the 
velocity  of  the  point  as  before  by  v ;  then  MN  =  vt' . 
The  arc  MN  being  infinitely  small,  the  direction  of 
the  accelerating  force  may  be  supposed  to  remain  par- 
allel to  MC  while  the  point  moves  from  M  to  N ;  and 
hence  during  the  time  t'  this  force  may  be  regarded 
as  constant  in  direction,  as  well  as  in  intensity.  If 
then  the  accelerating  force  were  to  act  alone  upon  the 
point  at  M,  it  would  cause  it  to  describe  the  Ime  MO 
during  the  time  t' ;  and  hence  denoting  the  force  by 
/,  we  shall  have  [Equa.  6] 

But  considering  the  arc  MN  as  coinciding  with  its 
chord,  we  have 

MO  X  MD  =  MN^ 

and  hence  M  0  =  ^' ; 

and  denoting  the  radius  MC  by  r,  and  substituting  for 
MN  its  value  vt' ,  we  have 

Hence  we  have         xff^^t^^ 

and  /=- •    [15] 

The  tension  of  the  string  to  which  the  force  /  is 
equal  and  opposite,  is  called  the  centrifugal  force,  and 
/  itself  is  called  the  central  or  centripetal  force ;  each 
having  for  its  measure  the  square  of  the  velocity  divided 
by  the  radius  of  the  circle. 


DYNAMICS.  129 

We  have  considered  the  radius  CM  constant;  but 
we  may  suppose  it  to  vary  in  length,  so  that  the  point 
shall  describe  any  curve  whatever.  In  that  case  the 
centrifugal  force  will  vary  from  one  point  to  another ; 
but  at  any  point  of  the  curve,  it  will  be  equal  to  the 
square  of  the  velocity  divided  by  the  radius  of  the 
osculating  circle.  For,  the  osculating  circle  at  any 
point  coinciding  for  an  infinitely  small  space  with  the 
curve  itself,  the  material  point  may  be  considered  as 
moving,  at  each  point  of  the  curve,  on  the  arc  of  the 
osculating  circle  at  that  point. 

24.  To  find  the  relation  between  gravity  and  the 
centrifugal  force,  when  the  material  point  revolves 
on  the  circumference  of  a  circle,  let  h  be  the  height 
from  which  a  body  must  fall  in  order  to  acquire  the 
velocity  v ;  then  v^  =  2gA,  and  substituting  the  value 
of  i;  in  j:he  equation  [15], 

J       r' 

we  find  Z.^?^. 

g        r 

That  is,  the  centrifugal  force  is  to  gravity,  as  twice  the 
height  due  to  the  velocity  of  the  material  point,  is  to  the 
radius  of  the  circumference  described  by  the  point. 

25.  If  we  denote  by  T  the  time  in  which  a  mate- 
rial point,  moving  uniformly  with  a  velocity  v,  will 
describe  the  entire  circumference  of  a  circle  of  which 
the  radius  is  r,  we  shall  have 

27rr 

and  substituting  this  value  of  v  in  the  equation 

17 


130  MECHANICS. 

J-  ^' 

we  get  f^'^'^^'Z, 

For  a  point  which  describes  a  circumference  of  which 
the  radius  is  r',  in  the  time  T",  we  in  like  manner  get 

fi  __  _"_!_• 

and  hence  we  have 

/:/'::     ^     :     ^, [16] 

That  is,  the  central  forces  in  the  two  cases  are  directly 
as  the  radii,  and  inversely  as  the  squares  of  the  times. 

26.  An  ilhistration  of  the  foregoing  results  is  fur- 
nished by  the  revolution  of  the  earth  on  its  axis. 
Assuming  the  figure  of  the  earth  to  be  that  of  a 
sphere,  let  PP'  [Fig.  14]  represent  its  axis,  and  the 
semi-circumference  PEP'  the  terrestrial  meridian  of 
any  place  M  on  its  surface.  From  the  centre  C  draw 
CE  perpendicular  to  PP',  and  draw  MN  parallel  to  it. 
Draw  also  CM.  Then  as  PEP'  revolves,  the  point  E 
will  describe  the  equator,  and  M  a  parallel  of  latitude ; 
and  as  each  circle  is  described  in  the  same  time, 
denoting  the  centrifugal  forces  at  E  and  M  by  /  and 
/',  and  the  latitude  of  the  place  by  n,  we  shall  have 
(proportion  16) 


/     :     /'     : 

:  :     EC 
:  :     EC 

:      MN, 

:     CM  .  cos  n, 

or 

/     :     /' 

:  :     1     : 

cos  n, 

and  hence 

Z'^/- 

cos  n.   . . 

■  ■ .  ["J 

DYNAMICS.  131 

That  is,  the  centrifugal  force  at  any  place  on  the  earth'' s 
surface,  is  equal  to  the  centrifugal  force  at  the  equator, 
multiplied  by  the  cosine  of  the  latitude  of  the  place. 

27.  The  effect  of  the  centrifugal  force  due  to  the 
rotation  of  the  earth,  is  evidently  to  diminish  the 
intensity  of  gravity  at  all  points  of  the  earth's  sur- 
face except  at  the  poles.  At  the  equator,  the  centri- 
fugal force  and  gravity  are  directly  opposed  to  each 
other.  Hence  if  we  denote  the  intensity  of  gravity 
at  the  equator,  as  determined  by  observation,  by  g ; 
the  intensity  which  it  would  have,  did  the  earth  not 
revolve  on  its  axis,  by  G,  and  the  centrifugal  force  at 
the  equator  by/,  we  shall  have 

The  value  of  g  has  been  found  to  be  equal  to 
32,^'"0861.  The  general  value  of/  [Art.  25],  is  given 
by  the  equation 

and  for  the  case  in  question,  we  have  r,  or  the  equa- 
torial radius  of  the  earth  =  20920300  feet;  and  T 
the  sidereal  day,  or  the  time  of  the  earth's  revolution 
on  its  axis  =  0,997269  of  a  mean  solar  day  =  86164' ' 
of  mean  solar  time ;  and,  hence,  by  substitution,  we  get 
/=0/t.lll2 [a] 

We  thus  have 

a  =  32,ft08()l  +  0,ftlll2  =  32/tl973 [b] 

28.  Dividing  equation  [a]  by  [b],  we  get 

f       0  ^'1112         1  1 

ii  =  327:1m    =oSg'   and  hence /=g-ggG; 


132  MECHANICS. 

that  is,  at  the  equator,  the  centrifugal  force  is  the  2^9^^ 
of  the  intensity  which  gravity  would  have,  had  the  earth 
no  motion  of  rotation. 

29.  At  any  point  of  the  earth's  surface,  not  on  the 
equator,  the  directions  of  gravity  and  the  centrifugal 
force  are  oblique  to  each  other.  Thus  at  M  [Fig.  14], 
gravity  acts  in  the  direction  CR,  and  the  centrifugal 
force  in  the  direction  NO.  To  determine  the  effect 
of  the  latter  in  the  direction  CR,  let  the  centrifugal 
force  be  represented  by  the  line  MO,  and  be  resolved 
into  the  components  MR,  MQ,  the  one  in  the  line  CR, 
the  other  at  right  angles  to  it ;  then  denoting  the  lati- 
tude of  M  as  before  by  n,  we  get 

MR  =  MO  .  cos  71. 
But  [equa.  17]  MO  =/'=/cos  ?i, 
and  hence  MR  =/  cos'  ?z ; 

that  is,  the  diminution  of  gravity,  due  to  the  centrifugal 
force  at  any  place  on  the  earth'' s  surface,  is  equal  to  the 
centrifugal  force  at  the  equator,  multiplied  by  the  square 
of  the  cosine  of  the  latitude  of  the  place. 

Experiments  with  the  pendulum  show  that  the 
intensity  of  gravity  at  the  poles  exceeds  the  intensity 
at  the  equator  by  2lolh  of  G;  of  this  increment,  i]\Q 
aigth  is,  as  we  have  seen,  due  to  the  centrifugal  force  ; 
the  remainder  is  accounted  for  by  the  spheroidal  figure 
of  the  earth. 

30.  As  the  centrifugal  force  depends  upon  the  time 
of  rotation,  it  may  be  proposed  to  determine  in  what 
time  the  earth  should  revolve  on  its  axis,  in  order  that 
the  centrifugal  force  at  the  equator  may  be  equal  to 


DYNAMICS.  133 

gravity.  For  the  solution  of  this  problem  we  have 
recourse  to  proj)ortion  [IG], 

T  I*' 

J        '•      J  *     *        2^       •        JTTa ' 

In  this  proportion  we  make  r  =  r^  =  the  equatorial 
radius  of  the  earth,  and  we  suppose  T  and  T '  to  rep- 
resent respectively  the  present  and  required  periods 
of  rotation,  and  fandf  ==  G  the  corresponding  cen- 
trifugal forces.     We  thus  get 

f     :     G     :   :      T"     :      T% 

and  hence  T'  =    \f^    T- 

or  T'  =  x/^i^  .  T=J^  .  T. 

Thus  the  required  time  is  \^th  of  the  present  period  of 

rotation. 


OF    PROJECTILES. 

31.  A  body  moving  near  the  surface  of  the  earth, 
in  virtue  of  a  primitive  impulse  and  the  action  of 
gravity,  is  called  a  projectile.  In  what  follows,  we 
shall  consider  the  projectile  as  reduced  to  a  material 
point,  and  the  motion  to  take  place  in  a  vacuum. 

Suppose  then  a  material  point,  situated  at  A  [Fig. 
15]  to  receive  an  impulse  in  the  direction  AD,  and 
then  to  be  abandoned  to  the  action  of  gravity.  Let 
the  space  which  it  would  describe  in  an  instant  in 
virtue  of  this  impulse  be  represented  by  AD,  and  let 
the  effect  of  gravity  during  the  instant  be  represented 


134  MECHANICS. 

by  AE  ;  then  at  the  end  of  the  first  instant,  the  point 
will  be  found  at  B,  having  during  the  instant  described 
the  diagonal  AB  of  the  parallelogram  constructed  on 
AE  and  AD. 

Again,  on  AB  produced,  take  BG  =  AB ;  and  on 
BG,  and  the  vertical  BF==  AE,  construct  the  paral- 
lelogram BFCG ;  at  the  end  of  the  second  instant,  the 
point  will  be  at  C,  having  described  the  diagonal  BC. 
In  like  manner,  during  the  third  instant,  the  point 
will  describe  the  side  CH ;  during  the  fourth,  the  side 
HI,  and  so  on.  But  each  diagonal  being  infinitely 
small,  the  series  of  diagonals  forms  a  curve ;  and  since 
each  of  the  parallelograms  has  its  contiguous  sides  in 
the  vertical  plane  which  contains  the  preceding  par- 
allelogram, all  the  points  of  this  curve  are  in  the  same 
vertical  plane.  This  curve  is  called  the  trajectory  of 
the  material  point. 

32.  To  find  the  equation  of  the  trajectory,  through 
the  point  A  [Fig.  16]  from  which  the  material  point 
is  projected,  draw  the  axes  AX,  AY,  horizontal  and 
vertical  respectively.  Let  AK  be  the  direction  of  the 
primitive  impulse,  and  denote  the  velocity  due  to  the 
impulse  by  v.  Let  the  curve  described  be  represented 
by  AMC,  and  let  M  be  the  position  of  the  point  at 
the  end  of  the  time  t.  Draw  MP  perpendicular  to 
AX,  and  produce  it  to  M' .  Put  AP  =  x,  PM  =  ?/, 
and  the  angle  M'  AP  =  i.  Then  let  the  initial  velo- 
city V,  represented  by  AM",  be  resolved  into  the 
components 

AP'  =  V  cos  2,  and  AQ  =  v  sin  i. 


I 


DYNAMICS.  135 

The  material  point  may  evidently  be  regarded  as 
having  two  motions,  the  one  parallel  to  AX,  the  other 
to  AY.  The  motion  parallel  to  AY  is  the  same  as 
that  of  a  body  projected  vertically  upwards  with  a 
velocity  v  sin  i  \  and  hence  we  have  [Art.  9] 

y  =  v  sin  i  t  —  \ge [18] 

The  other  is  due  to  the  horizontal  component  v  cos  i, 
and  gives 


or  t  = 


X  =  V  COS  i  t, 

X 


V  COS  e 

Substituting  this  value  of  t  in  equation  [18]  and  put- 
ting 2gh  in  place  of  v'^  [Art.  8,  equa.  9J  we  get 

4:hy  cos^  i  =  4/i.c  sin  i  cos  i  —  x'^ ; [19] 

and  hence 

X  =  2k  sin  i  cos  i  dz  ^/ih  CDs'*  i  {h  sin'^  i  —  y).    [20] 

33.  The  distance  from  the  origin  A  to  the  point  C, 
where  the  curve  intersects  the  horizontal  line  AX,  is 
technically  called  the  range.  To  determine  its  value, 
in  equation  [20],  we  make  2/  =  0  ;  we  thus  get 

AC  =  4A  sin  i  cos  i 
=  2A  2sin  i  cos  i 

--:  2A  sin  2z [21] 

This  is  the  general  expression  for  the  range. 
When  2  =  45°,  or  2/ =  90°,  we  have 
AC  =  2A. 

This  is  evidently  the  maximum  value  of  AC.     Hence, 
for  a  given  initial  velocity,  the  range  is  greatest  when 


136  MECHANICS. 

the  angle  of  projection  is  equal  to  45^.  Denoting  this 
value  of  the  range  by  R ,  we  have 

R'  =  '2h, 

and  hence  h  =  ?^'-\ 

from  which  it  appears  tliat  the  height  due  to  the  initial 
velocity  is  equal  to  half  the  maximum  range  Hence,  to 
determine  A,  we  have  only  to  measure  the  range  when 
the  angle  of  projection  is  45°. 

34.  Resuming  equation  [21]  and  denoting  AC  by 
R,  we  have 

R  =  2h  sin  2z ; 

and  substituting  for  2A  its  value  R ,  we  have 
R=R'  sin  22, 

an  equation  which  gives  the  range  corresponding  to 
any  angle  of  projection,  when  the  maximum  range  for 
the  same  initial  velocity  is  known. 

35.  Let  us  now  suppose  two  material  points  to  be 
projected  with  equal  initial  velocities,  and  denote  the 
angles  of  projection  by  i'  and  ^",  and  the  correspond- 
ing ranges  by  i?'"  and  R'''.     We  have,  equa.  [21], 

i^'"=2/i,sin2z', 
i^'^-  =  2Asin2i"; 

and  if  i' '  =  (90^  —  i' ),  we  get 

i?i^'-2Asin(180°  — 22') 
=  2h  sin  2i', 

and  R'"  =  i^iv . 

that  is,  whe7i  two  material  points  are  projected  with  equal 


DYNAMICS.  137 

initial  velocities^  at  angles  of  projection  which  are  com- 
plementary to  each  other ^  the  range  is  the  some  for  each. 

36.  From  equation  [20]  we  learn  that  the  greatest 
elevation  of  the  point,  or  the  maximum  value  of  y,  is 
h  sin~  i ;  for  if  y  be  greater  than  h  sin~  f,  the  radical 
becomes  imaginary.  The  value  of  x  corresponding  to 
the  maximum  value  of  y,  is  2h  sin  i  cos  i.  Supposing 
AE  and  ED  to  be  these  values,  we  have 

AE  ==  2k  sin  i  cos  i, 
ED  =  k  sin'  i. 

37.  If,  when  the  initial  velocity  is  given,  it  be 
required  to  determine  what  must  be  the  value  of  the 
angle  of  projection,  in  order  that  the  projectile  may 
reach  a  given  point  N,  for  which  we  have 

X  =  AQ'  =  x',  and  y  —  Q'N  =  y', 
we  have  recourse  to  equation  [19].     Substituting  in 
this  equation  the  given  values  of  x  and  y,  and  divid- 
ing each  member  by  cos^  i,  we  get 

Uy'  =  Ux'^'''^        ^" 


cos  I 


and  since         _!_,  =  sec' i  =  1  +  tang' z, 

COS" I  '  ° 

we  have        Uy'  =  Ux'  tang  i  —  x"  —  x"  tang'  i, 

or                     .       2-       4A,         .       —4ky'  —  x" 
or  tang'  z  —  -^,-  tang  z  = 1^^ ; 

and  hence         tang  i  ==  ^h±:s/4h^--4hy' -x'^ 

We  learn  from  this  equation,  that  in  order  that  the 
problem  may  be  possible,  we  must  have 

18 


138  MECHANICS. 

and  that  when  this  condition  is  satisfied,  there  are  two 
directions  in  tvhich  the  projectile  may  he  thrown,  so  as  to 
reach  the  given  point. 

38.  The  time  employed  by  the  projectile  in  describ- 
ing any  portion  of  the  trajectory,  may  be  found  from 
the  equation 

x=v  COS  i  t. 

39.  In  equation  [19]  the  points  of  the  curve  are 
referred  to  the  axes  AX,  AY.  Let  us  now  refer  them 
to  the  axes  DE  and  DY'  [Fig  17]  drawn  through  D, 
the  highest  point  of  the  trajectory,  parallel  to  AY  and 
AX  respectively.  Denote  DR,  the  new  abscissa  of 
the  point  M,  by  x^  ;  and  MH,  the  new  ordinate,  by  y'  : 
we  have  [Art.  36] 

a:  =  2A  sin  i  cos  i  —  y'^ 
and 

y  =  h  sin^  i  —  x'. 

Substituting  these  values  of  x  and  y  in  equation  [19] 

4:hy  cos'^  i  =  4:kx  sin  i  cos  i  —  x"^, 

we  have 

4:h  {k  sin"  i  —  x')  cos'^  i  =4:h  {2k  sin  i  cos  i  —  y')  sin  i  cos  i 

—  (2A  sin  i  cos  i  —  y'Y  ; 
or  reducing, 

4^^  sin'^  i  cos^  i  —  4A  cos^  i  x'  =  8A^  sin^  i  cos'^  i  —  ih  sin  i  cos  i  y' 
—  Ah"^  sin'^  i  cos*^  i  -f-  4A  sin  i  cos  i  y'  —  y'  '^, 

and,  by  a  farther  reduction, 

y'^  =4ih  QOS^  i  x'. 


DYNAMICS.  139 

From  this  equation,  it  appears  that  the  trajectory 
is  a  parabola,  having  its  vertex  at  D,  and  of  which  the 
parameter  to  the  axis  is  4h  cos^  i.  ^ 


MEASURE    OF    FORCES. 

40.  In  article  8  of  Part  First,  we  have  shown  how 
the  intensities  of  forces  may  be  represented  by  alge- 
braic symbols ;  and  assuming  the  forces  to  act  upon 
single  material  particles,  we  have  also  shown  that  the 
spaces  described  by  different  particles  in  the  same 
time,  or  their  velocities,  are  directly  proportional  to 
the  intensities  of  the  forces ;  and,  conversely,  that  the 
forces  are  directly  as  the  velocities.  Thus  denoting 
the  forces  which  act  separately  upon  two  material 
particles  by  /  and  /' ,  and  the  velocities  which  they 
communicate  to  the  particles  by  v  and  v' ,  we  have 
shown  that 

f     '.    f     :   '.     V     :     V' [a] 

1°.  Let  us  now  substitute  for  the  particles,  two 
bodies  of  equal  masses,  each  body  consisting  of  M  par- 
ticles ;  and  let  us  suppose  each  particle  of  the  first  to 
be  acted  upon  by  a  force  equal  to/,  and  each  particle 
of  the  second  by  a  force  equal  to  /'  ;  the  impulses 

*  That  the  trajectory  is  a  parabola,  may  also  be  shown  by  referring 
the  points  of  the  curve  to  the  obhque  co-ordinates  AX,  AK.  The 
method  in  the  text  is  employed,  because  it  illustrates  the  simplification 
which  frequently  results  from  a  change  of  co-ordinates. 


140  MECHANICS. 

being  supposed  to  be  given  in  the  same  sense,  and  in 
parallel  directions.  The  bodies  will  evidently  move 
with  the  velocities  v  and  v'  (all  their  particles  describ- 
ing parallel  straight  Imes) ;  and  their  motions  may  be 
considered  as  due  directly  to  the  resultants,  equal  res- 
pectively to  Mf  and  Mf ' ,  of  the  M  equal  and  parallel 
components,  which  act  upon  the  particles  of  each 
body.  Denoting  these  resultants  by  Q  and  Q' ,  we 
have 

Q  =  Mf,tindQ'  =  Mf'', 
and  hence 

Q     :     Q'     :  :     Mf    :     Mf 

or,  equa.  [a],       Q     :     Q'     :  :     v     :     v' .  .  .  [b] 

Thus  afiy  two  forces  are  to  each  other  as  the  velocities 
which  they  would  respectively  communicate  to  two  bodies 
of  the  same  mass ;  or,  more  briefly,  when  the  masses  are 
equal,  the  forces  are  as  the  velocities. 

2^.  Referring  again  to  the  two  bodies,  let  us  sup- 
pose them  to  have  unequal  masses,  the  one  to  have 
a  mass  m,  the  other  a  mass  M;  and  let  us  conceive 
each  of  their  particles  to  be  acted  upon  by  a  force 
equal  to/.  The  bodies  will  evidently  move  with  the 
common  velocity  v ;  and  their  motions  may  be  con- 
sidered as  due  to  the  resultants,  equal  to  mf  and  Mf 
respectively,  of  the  parallel  components  which  act 
upon  their  particles.  Denoting  these  resultants  by 
F  and  Q  respectively,  we  shall  have 

F     :     Q     :  :     7?if    :     Mf 

:  :      m     :     M^    [c] 


DYNAMICS.  141 

Thus  any  two  forces  are  to  each  other,  as  the  masses  to 
which^  if  respectively  applied,  they  would  communicate 
equal  velocities ;  or,  when  the  velocities  are  equal,  the 
forces  are  as  the  masses. 

3°.  Having  ascertained  the  relation  between  the 
intensities  of  two  forces,  1st,  When  the  forces  com- 
municate unequal  velocities  to  equal  masses,  and, 
2d,  When  they  communicate  equal  velocities  to  une- 
qual masses,  we  will  now  consider  the  general  case, 
that  in  which  the  forces  communicate  unequal  velo- 
cities to  unequal  masses. 

Let  F  and  F'  denote  the  forces,  m  and  m!  the 
masses,  and  v  and  v'  the  velocities  respectively.  Also 
let  Q  and  Q'  denote  two  auxiliary  forces,  whose  inten- 
sities are  such  that  they  would  communicate  to  two 
bodies  of  the  same  mass  M  the  velocities  v  and  v' 
respectively. 

We  have  [1^  of  this  Art.], 

q     :     Q'     :   :     V     :     V'. 

We  also  have  [2°  of  this  Art.], 

F     :     q     :  :     m     :     M, 
F'     :      Q'     :   :     7n'     :     M; 

and  eliminating  the  quantities  Q,  Q'  and  M,  we  get 

F     :     F'     :   :     mv     :     m'v' [dj 

The  product  of  the  mass  m  of  a  body  by  its  velo- 
city V,  is  called  the  quantity  of  motion  of  the  body. 
Employing  this  term,  the  proportion  just  found  may 
be  thus  enunciated : 


142  MECHANICS. 

Ann  tivo  forces  are  to  each  other  as  the  quantities  of 
motion  which  they  would  respectively  generate  during  the 
same  time. 

4°.  If  in  proportion  [d]  we  suppose  m'  to  become 
the  unit  of  mass,  and  v'  the  unit  of  velocity,  F'  will 
become  the  force  which  will  communicate  to  the  unit 
of  mass  a  velocity  equal  to  the  unit  of  velocity ;  and 
if  we  take  it  for  the  unit  of  force,  w^e  shall  have 

P  =  mv,    [e] 

mv  being  the  ratio  of  the  force  F  to  the  unit  of  force. 
5°.  If  in  proportion  [d]  we  suppose  F  =  F' ,  we  get 
fnv  =  m'v' ,  and  hence 

771'V'  ri    T 

V  = [h] 

Thus  the  quantity  of  motion  m'v'  due  to  a  force  F 
being  given,  to  find  the  velocity  which  the  same  force 
would  communicate  to  a  body  whose  mass  is  m,  we 
have  only  to  divide  the  given  quantity  of  motion  by 
this  mass. 

6°.  The  preceding  relations  are  evidently  true, 
whatever  the  nature  of  v  :  whether  it  be  a  finite  velo- 
city, due  to  a  force  of  the  kind  called  impulsive,  or  an 
infinitely  small  velocity,  such  as  we  conceive  a  force 
acting  at  infinitely  small  intervals,  like  gravity,  to 
communicate  at  the  commencement  of  each  interval ; 
or,  lastly,  whether  it  be  the  finite  velocity  produced 
by  the  latter  kind  of  force  (that  is,  an  accelerating 
force)  acting  during  a  given  time. 

Reserving  v  to  denote  the  velocity  due  to  an  impul- 
sive force,  and  employing  </>  to  denote  the  velocity  due 


DYNAMICS.  143 

to  an  accelerating  force  acting  uniformly  during  the 
unit  of  time,  we  shall  have,  for  the  two  cases, 

F  =  mv,    [k] 

F  =  7n(f) [k'] 

If  in  these  equations  we  make  m  =  1,  we  get 

F  =  v,     F=(P; 

and  F  becomes  in  each  case  the  force  which  acts  upon 
a  material  particle,  or  the  unit  of  mass.  For  the  sake 
of  brevity,  </>  is  commonly  called  the  accelerating  force ; 
and  mcf)  the  general  expression  of  Fin  equation  [k  ], 
is  called  the  moving  force. 

41.  Nature  of  an  impulsive  force. 

We  learn  from  experiment,  that  in  all  cases  of  col- 
lision, the  bodies  concerned  suffer  a  sensible  compres- 
sion, of  greater  or  less  extent,  according  to  their 
degree  of  hardness.  This  compression  is  evidently 
effected  in  a  finite,  though  very  short  time,  and  by 
infinitely  small  degrees.  Hence  in  the  production  of 
motion  by  an  impulsive  force,  we  may  conceive  the 
transmission  of  the  motion  from  the  one  body  to  the 
other  also  to  take  place  by  infinitely  small  degrees. 
Thus  in  this  case,  as  in  that  of  an  accelerating  or 
moving  force,  we  may  conceive  the  time  during  which 
the  force  acts,  to  be  divided  into  an  infinite  number 
of  instants,  and  suppose  an  infinitely  small  velocity 
to  be  communicated  at  the  commencement  of  each 
instant ;  the  essential  difference  in  the  two  cases  beino; 
that  the  infinitesimal  velocities  communicated  by  the 
impulsive  force  must  be  supposed  to  be  vastly  greater 


144  MECHANICS. 

than  those  communicated  by  the  movmg  force.  An 
impulsive  force  may  therefore  he  regarded  as  a  moving 
force,  acting  for  a  very  short  time,  with  a  very  great 
intensity.  Since  then  the  finite  impulse  given  by  an 
impulsive  force  may  be  supposed  to  be  made  up  of  an 
infinite  number  of  infinitely  small  impulses,  we  may 
simplify  our  first  notion  of  the  mode  of  action  of  a 
force  (Art.  2  of  Statics),  and  say  that  the  action  of  a 
force  may  always  be  conceived  to  consist  in  commu- 
nicating to  the  particles  of  the  body  on  which  it  acts, 
infinitely  small  impulses  of  greater  or  less  intensity. 

42.  To  give  a  simple  example  of  the  application  of 
some  of  the  preceding  results  (Art.  40),  let  m  and  m' 
denote  the  masses  of  two  bodies,  suspended  at  the 
extremities  of  a  cord  which  passes  over  a  fixed  pulley, 
and  let  the  velocity  which  gravity  will  communicate 
to  the  bodies  during  a  unit  of  time  be  denoted  by  g. 
The  moving  forces  put  in  operation  by  this  disposi- 
tion of  the  bodies  will  be  mg  and  m'  g',  and  supposing 
m^m\  their  resultant  will  be  mg  —  m' g.  This  re- 
sultant, acting  upon  the  sum  {7n-\-m')  of  the  two 
masses,  will  cause  the  one  body  to  descend,  the  other 
to  ascend ;  and  the  velocity  g'  which  it  will  commu- 
nicate to  the  system  in  the  unit  of  time,  will  be  given 
(5°  of  this  Art.)  by  the  equation 

,       m  —  m' 

cr    = or 

^       m  +  m'^ 

The  simple  arrangement  just  considered,  is  the 
skeleton  of  a  machine  called,  from  the  name  of  its 
inventor,  Atwood's  machine;  in  which,  by  rendering 


DYNAMICS.  145 

the  difference  between  the  masses  sufficiently  small, 
the  motion  of  the  bodies  is  rendered  so  slow  that  the 
velocity  acquired,  and  the  space  described,  in  any 
given  time,  can  be  determined  by  actual  observation. 
It  has  thus  been  found  that  bodies  near  the  surface 
of  the  earth  fall  with  a  uniformly  accelerated  motion, 
and  hence  that  terrestrial  gravity  is  a  constant  accele- 
rating force. 


PRINCIPLE    OF    D  ALEMBERT. 

43.  In  the  preceding  sections,  we  have  constantly 
supposed  the  body  whose  motion  was  the  subject  of 
investigation,  to  be  reduced  to  a  single  material  point. 
We  purpose  now  to  consider  some  of  the  more  elemen- 
tary cases  of  the  motion  of  systems  of  points,  or  bodies 
of  sensible  magnitude.  But  the  student  must  first  be 
made  acquainted  with  a  general  princij)le  of  great 
utility  in  the  solution  of  problems  of  dynamics, 
called,  from  the  name  of  its  discoverer,  the  principle 
of  D'  Alemhert. 

Let  A,  A',  A",  A'",  etc.,  be  a  system  of  material 
points,  connected  with  each  other  in  any  manner 
whatever,  and  acted  upon  by  the  accelerating  forces 
/,/',/",/'",  etc.  respectively.  The  velocities  which 
the  points  will  actually  acquire  in  an  infinitely  small 
time  or  instant,  under  the  action  of  these  forces,  will, 
in  consequence  of  their  mutual  connexion,  be  different 
in  both  intensity  and  direction  from  those  which  they 

19 


146  MECHANICS. 

would  acquire  were  they  free.  Thus  if  the  velocity 
which  /  would  communicate  to  A  [Fig.  18],  in  an 
instant  if  that  j^oint  were  free,  be  represented  by  AC, 
the  velocity  actually  acquired  by  A  will,  in  conse- 
quence of  the  connexion  of  the  points  of  the  system 
be  different  from  AC  and  may  be  represented  by  some 
other  line  AD.  Let  the  velocity  AD  be  considered 
a  component  of  AC,  and  let  the  parallelogram  ABCD 
be  completed:  we  shall  then  have  the  velocity  AC 
resolved  into  the  two  velocities  AD  and  AB.  To  dis- 
tinguish these  velocities  from  each  other,  we  call  AC 
the  impressed  velocity,  AD  the  effective  velocity,  and  AB 
the  velocity  lost.  Let  the  velocity  which  /  would 
communicate  to  A  in  the  unit  of  time  if  it  were  free, 
be  denoted  by  u ;  then  the  velocity  which  it  would 
communicate  in  the  instant  V ,  or  the  impressed  velo- 
cit}^,  will  be  expressed  by  uV  ;  and  if  we  denote  the 
components  of  u  by  q  and^,  the  effective  velocity  and 
the  velocity  lost  will  be  expressed  by  qt'  and  pt'  res- 
pectively. If,  moreover,  we  denote  the  mass  of  the 
point  A  by  m,  then  for  this  point 

the  impressed  quantity  of  motion  will  be  expressed  by  mut' ; 

the  effective  quantity  of  motion  by viqt', 

and  the  quantity  of  motion  lost  by mpt'. 

If  for  the  points  A' ,  A" ,  A" ' ,  etc.,  we  denote  the 
quantities  corresponding  to  u,  q,  p  and  m,  by  these  let- 
ters, accented,  that  is,  hy  u' ,  q' ,  p' ,  m'  ;  u" ,  q" ,  p" , 
??i" ,  etc.,  the  several  quantities  of  motion  for  the 
entire  system  will  be  thus  expressed : 


DYNAMICS.  147 

The  impressed  quantities  of  motion  by 

miLt\     m'u't\     vi"u"t',  etc. ; [a] 

the  effective  quantities  of  motion  by 

mqt\     7n'q't',     m"q"t\  etc. ;    [bj 

the  quantities  of  motion  lost  by  m 

mpf,     m'p't',     m"p"t\  etc.  ; [c] 

Now  since  the  impressed  quantities  of  motion  are 
reduced,  in  consequence  of  the  mutual  connexion  of 
the  points,  to  the  effective  quantities  of  motion,  it  is 
obvious  that  the  quantities  of  motion  lost  must  he  in  equi- 
librium among  themselves.  In  this  consists  the  prin- 
ciple of  D'Alembert.  The  manner  in  which  the 
equilibrium  will  take  place,  will  of  course  depend 
upon  the  nature  of  the  system.    . 

44.  This  principle  may  also  be  enunciated  in  terms 
of  the  impressed  and  effective  quantities  of  motion. 
To  get  this  latter  enunciation,  produce  DA :  and 
taking  AD'  equal  to  AD,  complete  the  parallelogram 
ACBD' .  We  perceive  that  AB  the  velocity  lost  is 
the  resultant  of  AC  the  impressed  velocity,  and  AD' 
the  effective  velocity  taken  contrary  to  the  actual 
direction  of  the  motion.  Thus  for  the  point  A,  the 
quantity  of  motion  lost  may  be  resolved  into  the  im- 
pressed quantity  of  motion,  and  the  effective  quantity 
of  motion,  the  latter  being  taken  contrary  to  its  actual 
direction.  The  same  being  true  for  all  the  points  of 
the  system,  we  may  substitute  for  the  quantities  of 
motion  [c],  the  quantities  of  motion  [a]  and  [b],  the 
latter  [b]  being  taken  with  the  above  modification. 
But  the  quantities  of  motion  [c]  are  in  equilibrium 


148  MECHANICS. 

among  themselves :  consequently  an  equilibrium  must 
also  exist  between  the  quantities  of  motion  [a]  and 
[b],  the  latter  being  taken  contrary  to  the  actual 
motions.  Thus,  in  any  system  ivhatever,  there  will  be 
an  equilibrium  between  the  impressed  and  effective  quan- 
tities of  motion,  the  latter  being  taken  contrary  to  the 
actual  motions ;  regard  being  had,  in  forming  the  equa- 
tions of  equilibrium,  to  the  nature  of  the  system. 

This  enunciation  might  have  been  obtained  inde- 
pendently of  the  former ;  for  the  truth  of  it  is  obvi- 
ous, the  moment  the  terms  impressed  and  effective  quan 
titles  of  motion  are  understood  in  the  sense  in  which 
they  are  here  used.  The  preceding  method  has  been 
employed  to  insure  a  greater  familiarity  with  the  prin- 
ciple than  could  be  acquired  by  regarding  it  from  a 
single  point  of  view. 

We  have  supposed  the  particles  of  the  system  to  be 
acted  upon  by  accelerating  forces  only,  but  the  prin- 
ciple is  equally  true  when  the  forces  are  impulsive. 


MOMENT    OF    INERTIA. 

45.  The  sum  of  the  products  obtained  by  multi- 
plying the  masses  of  the  particles  of  a  body  by  the 
squares  of  their  respective  distances  from  any  line 
whatever,  is  called  the  moment  of  inertia  of  the  body 
with  respect  to  the  line. 

The  moment  of  inertia  of  a  body  is  represented 


DYNAMICS.  149 

algebraically  by  the  expression  i(r?n) ;  in  which  m 
denotes  the  mass  of  a  particle  of  the  body,  and  r  the 
distance  of  the  particle  from  the  axis  with  respect  to 
which  the  moment  is  taken.  When  the  moment  of 
inertia  of  a  body  with  respect  to  an  axis  which  passes 
through  its  centre  of  gravity  is  known,  its  moment  of 
inertia  with  respect  to  any  other  axis  can  be  readily 
determined. 

Let  C  [Fig.  19]  be  the  centre  of  gravity  of  the 
body,  and  FF'  an  axis  passing  through  it,  with  respect 
to  which  the  moment  of  inertia  of  the  body  is  known : 
and  let  its  moment  of  inertia  with  respect  to  any 
other  axis  KK',  parallel  to  FF',  be  required.  Let 
the  point  C  be  taken  for  the  origin  of  co-ordinates, 
and  the  axis  FF'  for  the  axis  of  z ;  and  let  CX  and 
CY  the  axes  of  x  and  y  be  drawn.  Let  N  be  the 
place  of  any  particle  of  the  body,  and  draw  through 
it  the  plane  NKF  parallel  to  the  plane  of  xy,  and 
meeting  the  axes  of  moments  in  the  points  F  and  K. 
Draw  NE  perpendicular  to  the  plane  of  xy,  EP  and 
GD  perpendicular  to  the  axis  of  x,  and  ED'  parallel 
to  that  axis ;  and  join  the  points  C  and  E,  C  and  G, 
E  and  G.  Let  CD  and  DG,  the  co-orHinates  of  G,  be 
denoted  by  a  and  /3 ;  CP  and  PE  the  co-ordinates  of 
E,  by  X  and?/;  and  CG,  the  distance  between  the 
two  axes,  by  a.  Also  let  NK=EG  be  denoted  by 
r,  and  NF  =  EC  by  r,.     We  have 

CG^  =  CD^  +  DG^  and  CE^  =  CP^  +  PE^ ; 
^r  a^  =  a^  +  fi\^ndr;^x^  +  f [a] 

We  also  have 


150  MECHANICS. 

or     '  r'={^x  —  a.Y-^{{i  —  yY 

or,  reducing  by  equations  [a], 

7-^  :^  r;  —  IcLx  —  2,(3z/  +  a" ; 
and  multiplying  each  member  of  this  equation  by  the 
mass  m  of  the  particle,  we  get 

r^m  =  r^m  —  laxm  —  2/3z/;?z  -|-  a^m. 

If,  for  other  particles  of  the  system,  we  denote  the 
quantities  corresponding  to  r,  r  ^^  x  and  ?/,  by  these 
letters  accented,  we  shall  have 

r'  ^/n  =  Tii^'ni  —  2ax'm  —  2(3y'??i  -f-  a'^m, 
r"  "^m  =  r,,?in  —  "2ax"m  —  2j3y"m  -\-  a'^?n, 
etc.  ; 

and  adding  these  equations,  supposed  to  extend  to  all 
the  particles  of  the  body,  we  shall  get 

l{r'7n)  =  l{r,'m)  —  2oil{xm)  —  2(3l{ym)  +  a""^??!. 

Now  i{xjn)  is  the  sum  of  the  moments  of  the  masses 
of  the  particles  with  respect  to  the  plane  of  yz,  which 
passes  through  the  centre  of  gravity  of  the  l)ody ;  and 
hence  we  have  i{xm)  =  0.  For  a  similar  reason,  we 
have  i{y??i)  =  0.  Also,  im  expresses  the  sum  of  the 
masses  of  all  the  particles  of  the  body,  or  its  entire 
mass  M.     Consequently  we  have 

l{r'm)  =  ^r;m)  -\-a\M.    [h] 

In  this  equation,  the  term  z{r'^hn)  expresses  the 
moment  of  inertia  of  the  body  with  respect  to  the 
axis  FF'  which  passes  through  its  centre  of  gravity ; 
and  i{r'^m)  expresses  its  moment  of  inertia  with  respect 


DYNAMICS.  151 

to  the  axis  KK'  parallel  to  the  first  axis,  and  at  a  dis- 
tance from  it  equal  to  a.  Hence  the  moment  of  inertia 
of  a  body  with  respect  to  any  axis,  is  equal  to  its  moment 
of  inertia  with  respect  to  an  axis  passing  through  its 
centre  of  gravity,  parallel  to  the  first,  plus  the  product  of 
the  ?nass  of  the  body  by  the  square  of  the  distance  between 
the  two  axes. 

The  second  member  of  equation  [b]  may  be  put 
under  the  form 

Hence  we  have 

or,  denoting 

I  l{r'7n)  =  M  {k' +  a\) [c] 

46.  The  determination  of  the  moments  of  inertia 
of  particular  bodies  requires  in  general  the  use  of  the 
integral  calculus,  but  in  many  cases  they  can  be  found 
by  more  elementary  processes.  We  will  give  a  few 
examples. 

1°.   The  straight  line. 

Let  it  be  required  to  find  the  moment  of  inertia 
of  a  material  straight  line,  with  respect  to  an  axis 
which  passes  through  the  centre  of  gravity  of  the  line, 
at  right  angles  to  it. 

Let  AB  [Fig  20]  be  the  line,  and  C  its  centre  of 
gravity ;  and  let  it  be  supposed  to  be  divided  into 
2n  indefinitely  small  equal  parts  or  elements,  by  sec- 


152  MECHANICS. 

tions  perpendicular  to  its  length.  Let  its  length  be 
denoted  bj  /,  and  the  area  of  a  section  by  s.     The 

volume  of  an  element  will  be  expressed  by  ^^j  ?  ^^^^ 
as  the  mass  is  here  supposed  to  be  proportional  to 
the  volume,  we  may  also  take  this  expression  as  the 
measure  of  the  mass  of  the  element.  Moreover  the 
respective  distances  of  the  elements  of  BC  (half  of 
the  line)  from  C,  that  is,  from  the  axis  of  moments, 
will  be  expressed  by 

— ,  2    ^  ,  3    ~,                 ....     71    -. 
2?i    ■    'In        'In     2?j 

Consequently  the  moment  of  mertia  of  BC  will  be 

2;^•    \ln)'^%i'    \ln)  ^ '^ 'In     \2/J  ' 

°^  T -^13+2 +6;- 

*  To  find  the  sum  of  the  series 

12  _|_  2^  _^  3^  -I-  n^ 

in  the  identical  equation 

(x-l)^-:(x-l)^ 
or  (x  — l)'  =  x-'  — 3x^4-3x  — 1, 

substitute  successively  for  x  each  term  of  the  series 

1,    2,    3,    4,   7j. 

We  shall  thus  get  the  following  equations  : 
0  =  1'  — 3  .  1^  +  3  .1  —  1, 
13  =  2^-3.  2" -1-3.  2  —  1, 
2='  =  3='  — 3  .3'-|-3  .  3  — 1; 

) 

(;i_2)'=(/i  — 1)'— 3(n  — ir-|-3(n  — D— 1, 
(n  — 1)3  =  71='— 3n^-f3n  —  l; 
and  if  we  add  the  corresponding  terms,  cancelling  those  which  are 
common,  and  denoting  the  sum  of  the  series  l-}-2-[~^~}~^«  •  •  •-j-'^ 


DYNAMICS.  '  153 


or  (since  we  must  suppose  tz  =  go  ) 


24 


Hence  the  moment  of  inertia  of  the  entire  line  will 

12 


be  expressed  by  ^ ;  or,  denoting  the  mass  of  the  line 


by  M,  by  M^;. 

The  moment  of  inertia  of  AB  with  respect  to  any 
other  axis  parallel  to  the  first,  and  at  a  distance  from 
it  denoted  by  a,  will  be  expressed  [Art.  45]  by 

2°.   The  rectangle. 

Let  it  be  required  to  determine  the  moment  of 
inertia  of  an  infinitely  thin  rectangular  plate. 

1st.  Let  ABDE  [Fig.  21]  be  the  plate,  and  let  the 
axis  HI  with  respect  to  which  the  moment  of  inertia 
is  to  be  taken,  be  supposed  to  pass  through  C  the 
centre  of  gravity  of  the  plate,  and  to  be  parallel  to  one 
of  its  sides,  as  AB.  Conceive  the  plate  to  be  made 
up  of  material  lines  perpendicular  to  HI.  The  mo- 
ment of  inertia  of  each  line  with  respect  to  this  axis 

by  5',  and  that  of  the  series  1^+2^+3^+4^ +7i'  by  s",  we  shall 

have 


and  hence 


=  T  +  (''+»T-T 

___n^_i_  n^i     n 


20 


154  MECHANICS 

will  be  expressed  by  Af^o ,  ^  denoting  tne  mass  of 
the  line  and  /its  length.  Hence  if  we  suppose  M  to 
denote  the  sum  of  the  masses  of  the  lines,  or  the  mass 
of  the   entire  plate,  the  moment  of  inertia  required 

will  be  expressed  by  Mp. 

2d.  Let  the  axis  be  supposed  to  pass  through  C,  at 
right  angles  to  the  surface  of  the  plate.  As  in  the  pre- 
ceding case,  conceive  the  plane  to  be  made  up  of 
material  lines  or  elements  parallel  to  the  side  AE : 
and  let  the  number  of  these  elements,  or  the  indefi- 
nitely small  equal  parts  into  which  Ave  suppose  the 
line  AB  to  be  divided,  be  denoted  by  2n.  Let  AB 
and  AE  be  denoted  by  b  and  /  respectively,  and  the 
tiiickness  of  the  plate  by  i.     The  volume,  and  hence 

the  mass  of  each  element,  will  be  measured  by  ^^; 

and  the  moment  of  inertia  of  an  element  will  be  ex- 
pressed by 

bU\  2n  '  12    ,    /  ±Y  I 

'2n  ]  —blj—  ~^\P  ■  2?i)    \  ' 

I       2^  J  . 

p  being  any  number  of  the  natural  series.  Hence 
the  sum  of  the  moments  of  inertia  of  all  the  lines  com- 
2)osing  the  semi-rectangle  ALKE  will  be 

bli/l    .    .b^^\     ,    bJl/r         /2bs^\  bli/r-     ,    mbs^\ 

2n.  \12    '    ^2/1^  /    '    2n  \12  "•    ^2n^  /  ~^  "     '  2/i  \12    •"  ^2^^  /  ' 

2n  \I2"    '    ^2n^    '  ^"3    ■"  2  "^  6"^  ' ' 


DYNAMICS.  155 

QY  Mi  (  l"^     xh"^         1       (n^  _L  ^'^  _i_  ^^"j  \ 

2  \  12  "^  T  •    n'    •  ^¥    •"  2~  "^  6V  ' 

or  i////M-A^\ 

2  A      12      /• 

Consequently  the  moment  of  inertia  of  the  entire 
rectangle,  its  mass  being  denoted  by  M  and  its  diag- 
onal by  d,  will  be  expressed  by  M  ^^. 

3°.  The  rectangular  parallelopipedon  and  triangular 
prism. 

Since  a  rectangular  parallelopipedon  [Fig,  22]  may 
be  supposed  to  be  made  up  of  a  series  of  equal  rec- 
tangular plates,  its  moment  of  inertia  with  respect  to 
an  axis  CC  passing  through  the  centres  of  gravity  of 
any  two  of  its  opposite  faces  will  be  expressed  by  the 

formula  Mp^ ;  M  being  supposed  to  denote  the  mass 

of  the  solid,  and  d  the  diagonal  of  either  of  the  faces 
to  which  the  axis  is  perpendicular. 

The  same  formula  also  expresses  the  moment  of 
inertia  of  each  of  the  equal  triangular  prisms  into 
which  the  parallelopipedon  is  divided  by  the  diagonal 
plane  ADD' A',  with  respect  to  the  axis  CC  which 
joins  the  middle  points  of  the  hypothenuses  AD  and 
A'  D'  of  their  triangular  bases ;  M  being  supposed  to 
denote  the  mass  of  either  of  the  two  prisms,  and  d  the 
hypothenuse  AD  or  A'D' . 


156  MECHANICS. 

THE    COMPOUND    PENDULUM. 

47.  The  simple  pendulum  has  only  an  ideal  exist- 
ence. It  is  essential,  therefore,  to  the  practical  appli- 
cation of  the  results  obtained  in  articles  19  and  20, 
relative  to  this  pendulum,  that  the  relations  between 
it  and  the  compound  pendulum  should  be  known. 
These  relations  we  are  now  to  investigate. 

Let  a  compound  pendulum  be  conceived  to  oscil- 
late about  a  horizontal  axis.  During  its  motion,  the 
material  points,  or  particles  of  which  it  is  composed, 
will  describe  arcs  of  circles,  the  planes  of  which  will 
be  perpendicular  to  the  axis.  Let  OGFH  [Fig.  23] 
be  one  of  these  planes,  and  0  the  point  in  which  it  is 
intersected  by  the  axis ;  the  axis  being  supposed  to 
be  perpendicular  to  the  plane  of  the  paper. 

The  jDendulum  being  in  motion,  suppose  that  at  the 
expiration  of  the  time  t^  the  particle  which  describes 
the  arc  HFG  has  arrived  at  the  point  A,  having  dur- 
ing that  time  described  the  arc  HA.  Denote  by  w 
the  angular  velocity  of  the  whole  system  at  this 
instant ;  that  is,  the  velocity  of  any  point,  as  I,  at 
the  distance  from  the  axis  which  we  assume  as  the 
linear  unit ;  and  denote  the  distance  0 A  by  r ;  then 
the  absolute  velocity  of  the  element  at  A  will  be 
expressed  by  rw.  In  the  following  instant,  this  velo- 
city will  be  increased  by  the  action  of  gravity.  Let 
g  the  intensity  of  gravity  be  represented  by  the  ver- 
tical line  AD,  and  be  resolved  into  the  two  compo- 
nents AE  and  AB,  the  one  in  the  direction  of  OA, 
and  the  other  perpendicular  to  it.     Ttie  first  being 


DYNAMICS.  157 

destroyed  by  the  re-action  of  the  fixed  point  C,  we 
have  to  consider  only  the  latter.  This  component 
causes  the  angular  velocity  to  vary,  and,  if  we  repre- 
sent the  angle  BAD  by  (5,  will  be  expressed  by  g  cos  6. 
If  we  denote  the  increment  which  the  angular  velo- 
city receives  during  the  instant  t' ,  by  w' ,  the  corre- 
sponding increment  of  the  velocity  of  the  particle  at  A 
will  be  expressed  by  rj  :  and  at  the  end  of  the  time 
t  +  t' ,  the  velocity  of  the  particle  will  be  expressed 
by  roj  +  Vgj'  . 

If  the  element  at  A  were  unconnected  with  the 
other  particles  of  the  system,  the  accelerating  force 
g  cos  6  would  communicate  to  it,  during  the  instant 
/' ,  the  velocity  g  cos  d  t'  ;  and  at  the  end  of  the  time 
t  +  t' ,  its  velocity  would  be  rco-\-g  cos  6  t'  ;  the  direc- 
tion of  the  velocity  being  AB,  the  same  as  at  the  end 
of  the  time  t. 

Now  these  increments  of  velocity  g  cos  6  t'  and  roj' 
are  what  we  have  called,  in  Art.  43,  impressed  and 
effective  velocities  respectively.  Hence  denoting  the 
mass  of  the  particle  at  A  by  m,  the  impressed  quan- 
tity of  motion  of  this  particle  will  be  g  cos  6  V  m,  and 
its  effective  quantity  of  motion  tg)  m.  All  that  we 
have  shown  to  be  true  of  the  particle  at  A,  being 
equally  true  of  all  the  other  particles  of  the  system, 
we  shall  have  for  the  impressed  quantities  of  motion 
of  the  whole  system  a  series  of  terms  each  of  the  form 
g  cos  6  t'm,  the  sum  of  which,  employing  the  usual 
symbol,  may  be  expressed  by 

l{g  C08  6  t'm). 


158  MECHANICS. 

We  shall  also  have  for  the  effective  quantity  of  motion 
of  the  system  a  series  of  terms  of  the  form  rj  m^  the 
sum  of  which  may  be  expressed  by 

Jj{rG)'m). 

Now,  according  to  the  princii3le  of  D'Alembert 
[Art.  44],  these  two  quantities  of  motion  will  be  in 
equilibrium  with  each  other,  if  the  latter  be  taken 
in  directions  contrary  to  the  actual  directions  of  the 
motions.  The  case  then  is  exactly  similar  to  that  of 
Art.  40  of  Statics :  and  the  condition  of  equilibrium 
is,  that  the  sum  of  the  moments  of  the  forces  which 
tend  to  turn  the  system  in  one  sense  about  the  fixed 
axis,  must  be  equal  to  the  sum  of  the  moments  of  the 
forces  which  tend  to  turn  the  system  in  the  opposite 
sense,  the  moments  being  taken  with  respect  to  the 
axis  itself. 

To  obtam  the  moments  of  the  forces  or  quantities 
of  motion,  we  have  only  to  multiply  the  quantities  of 
motion  (impressed  and  effective)  of  each  element,  by 
the  distance  of  the  element  from  the  axis.  We  shall 
thus  get  for  the  equation  of  equilibrium, 

2(r-w';??)  =  ^{rg  cos  6  t'm)  ; 

or  since  J  is  common  to  all  the  terms  of  the  first 
member,  and  t'  and  g  to  those  of  the  second, 

(jd''^{r^ni)  =  gf^ir  cos  6  m) ; 

and  hence  we  shall  have 

gV   _      l{r  cos  6  ?}i) 

Conceive  now  a  vertical  plane  to  be  drawn  through 


DYNAMICS.  159 

the  fixed  axis,  and  perpendiculars  to  be  drawn  to  it 
from  all  the  particles  of  the  pendulum.  Let  AN  =  y 
be  the  perpendicular  drawn  from  the  particle  at  A ; 
then  we  shall  have 

y  —  r  cos  6 ; 

and  hence,  substituting  y  in  place  of  r  cos  6  in  the  pre- 
ceding equation,  we  shall  get 

Gy'  _       nijm) 

The  expression  ^{ym)  represents  the  sum  of  the 
moments  of  all  the  particles  of  the  pendulum  with 
respect  to  the  vertical  plane  drawn  through  the  axis ; 
and  if  we  denote  by  y'  the  perpendicular  drawn  from 
the  centre  of  gravity  of  the  pendulum  to  this  plane, 
and  by  M  the  entire  mass  of  the  pendulum,  we  shall 
have  [Statics,  Art.  45], 

and  hence 

^'  y'^  run 

-f—^^W^y  ■•••^^] 

Let  C  [Fig.  24]  be  the  centre  of  gravity  of  the 
pendulum,  and  O'G'F'H'  the  plane  drawn  through  it 
perpendicular  to  the  axis,  and  let  0'  be  the  point  of 
intersection  of  the  plane  and  axis;  then  the  line  CN', 
drawn  perpendicular  to  the  vertical  O'F',  will  be  the 
line  which  we  have  denoted  by  y'  ;  and  denoting  O'C 
by  a,  and  the  angle  O'CN'  by  6' ,  we  shall  have 
y'  =  a  cos  6'. 

We  also  have  [Art.  45], 


160  MECHANICS. 

Hence,  by  substitution,  equation  [b]  will  become 

b)'  _  ga  cos  6' M  _  ga  cos  6' 

The  second  member  of  this  equation  is  obviously  [Art. 
10]  the  expression  of  the  angular  accelerating  force 
for  the  whole  pendulum. 

Let  us  now  seek  the  expression  of  the  angular 
accelerating  force  for  a  single  particle,  situated  at  R 
on  0'  C  produced ;  0'  R  being  supposed  to  be  without 
mass,  and  inextensible  and  inflexible,  so  that  the  line 
and  particle  together  constitute  a  simple  pendulum. 

On  the  supposition  that  the  compound  pendulum  is 
reduced  to  a  single  particle,  equation  [a]  becomes 

G)'  _  gym  . 
t'         f'm  ' 

and  applying  this  result  to  the  particle  at  R,  and 
denoting  the  line  0'  R  by  /,  we  get 

0)'  _  gl  COS  6' 

_  g  cos  6' 
I       ' 

The  second  member  of  this  equation  is  the  expression 
required. 

'  48.  If  now  it  be  proposed  to  determine  the  value 
that  must  be  given  to  /,  in  order  that  the  simple  pen- 
dulum (or  the  point  at  R)  may  oscillate  in  the  same 
time  as  the  compound  pendulum,  we  have  only  to  put 
the  expressions  of  the  accelerating  forces  in  the  two 
cases  equal  to  each  other.     We  thus  get 

ga  cos  6'  _g  cos  <5' , 
"^^  +  A;'  I      ■  ' 


DYNAMICS, 

and  hence 

a               1 

a^  +  A:-^           I  ' 

and 

a 

='^+1 

161 


49.  To  apply  this  formula  to  a  particular  case,  we 
must  determine  the  mass  M  of  the  given  pendulum, 
and  the  distance  a  of  its  centre  of  gravity  from  the 
axis  of  suspension.  We  must  also  calculate  the  mo- 
ment of  inertia  of  this  mass,  with  respect  to  the  axis 
passing  through  the  centre  of  gravity  parallel  to  the 
axis  of  suspension.  Then  dividing  this  moment  by 
the  mass,  we  shall  have  the  value  of  k~ ;  and  substi- 
tuting the  values  of  a  and  F  in  the  second  member  of 
the  equation,  we  shall  have  the  length  /  of  the  simple 
pendulum  which  will  oscillate  in  the  same  time  with 
the  compound  pendulum. 

If  through  R,  considered  as  a  point  of  the  compound 
pendulum,  a  line  be  drawn  parallel  to  the  axis  of  sus- 
pension, all  the  points  of  the  line  will  obviously  oscil- 
late in  the  same  manner  as  the  point  R.  This  line  is 
called  the  axis  of  oscillation  of  the  pendulum,  and  the 
points  0'  and  R  are  called  the  centres  of  suspension 
and  oscillation  respectively. 

60.  The  axes  of  oscillation  and  suspension  are  recip- 
rocal ;  that  is,  when  the  axis  of  oscillation  is  made  the 
axis  of  suspension,  the  latter  becomes  the  axis  of  oscil- 
lation. 

For  let  AB  [Fig.  25]  be  the  section  of  a  compound 
pendulum  by  a  plane  passing  through  its  centre  of 

21 


162  MECHANICS. 

gravity,  perpendicular  to  the  axis  of  suspension ;  and 
suppose  C  to  be  the  centre  of  gravity,  and  O'the  cen- 
tre of  suspension.  To  determine  the  length  of  the 
corresponding  simple  pendulum,  we  have  the  equation 

or,  since  ^5  =  0'  C, 

If  then  on  0'  C  produced,  we  lay  off  CR  equal  to  qtq' 

the  point  H  will  be  the  centre  of  oscillation. 

Now  let  the  pendulum  be  inverted,  and  the  point 
R  [Fig.  26]  be  taken  for  the  centre  of  suspension; 
then  denoting  the  length  of  the  corresponding  simple 
pendulum  by  /' ,  and  the  distance  between  the  centres 
of  gravity  and  suspension  by  «',  we  have 


l'  =  a'  + 


=  ^^+CR' 


or,  since  CR  =  — , 

O'G  ' 

O'G  ^   k^ 
O'G 

=  0'C  +  ^. 

Hence  l  =  i'. 

Thus  the  lengths  of  the  corresponding  simple  pendu- 
lums are  the  same,  and  the  oscillations  about  the  axes 
of  suspension  and  oscillation  are  made  in  equal  times. 


DYNAMICS.  163 

51.  Conversely,  if  the  duration  of  the  oscillations  of 
a  compound  pendulum  is  the  same  about  two  parallel  axes 
which  lie  in  the  same  plane  with  the  centre  of  gravity  of 
the  pendulum^  and  at  unequal  distances  from  that  point, 
the  distance  hetiveen  the  axes  will  he  equal  to  the  length 
of  the  simple  pendulum  which  will  oscillate  in  the  same 
time. 

For  representing  by  AB  [Fig.  27]  the  section  of 
the  pendulum  by  a  plane  which  passes  through  C  its 
centre  of  gravity,  and  supposing  the  points  0'  and 
0^  to  be  made  the  centres  of  suspension  alternately, 
we  have  for  /  and  /',  the  lengths  of  the  corresponding 
simple  pendulum, 

1=^  a-\-  —  aud  V  =  a'  -\ p , 

a  a 

a  and  a'  denoting  the  distances  O'C  and  O^C  respec- 
tively. But  if  the  oscillations  of  the  two  pendulums 
are  made  in  the  same  time,  we  shall  have 

—  =  «'  +  —,; 

a  a 

from  which  we  get  for  a'  the  values  a  and  -.     But 

O'O^  the  distance  between  the  axes  is  equal  to  the 
sum  of  a  and  a' ',  hence  we  have 

0'0,  =  a  +  a'; 

or,  employing  the  latter  value  of  «', 

a 

Hence  0'  0  ^  is  equal  to  the  length  of  the  simple  pen- 
dulum which  oscillates  in  the  same  time  as  the  com- 
pound pendulum. 


164  MECHANICS. 

OF    THE    COLLISION    OF    BODIES. 

52.  We  learn  from  experiment,  1°,  that  all  bodies 
in  nature  are  more  or  less  compressible ;  and,  2°,  that 
when  in  any  case  a  body  suffers  compression,  it  tends 
in  a  greater  or  less  degree  to  resume  its  primitive 
figure.  The  tendency  of  a  body,  when  compressed, 
to  resume  its  primitive  figure,  is  called  its  elasticity. 
The  elasticity  of  a  body  varies  with  the  nature  of  its 
substance.  The  limiting  cases  of  the  elasticity  of 
bodies  are,  1st,  the  case  in  which  the  body  tends  to 
resume  exactly  its  primitive  figure ;  and,  2d,  that  in 
which  it  has  no  tendency  in  any  degree  to  recover 
its  figure.  In  the  first  case  the  body  is  said  to  be 
perfectly  elastic ;  in  the  second,  to  be  inelastic.  Neither 
of  these  cases  is  found  in  nature  :  no  known  substance 
being  either  jDcrfectly  elastic,  or  entirely  destitute  of 
elasticity.  All  known  substances  are  hence  said  to 
be  imperfectly  elastic.  There  are,  however,  some  sub- 
stances in  which  the  elasticity  approaches  very  near 
to  the  extreme  cases. 

53.  Inelastic  bodies. 

Let  A  and  B  [Fig.  28],  be  two  inelastic  homoge- 
neous bodies  of  spherical  form,  movmg  from  left  to 
right  on  the  straight  line  A'  B'  which  joins  the  cen- 
tres C  and  C  ;  and  let  A  be  supposed  to  have  the 
greater  velocity.  When  A  overtakes  B,  a  mutual 
compression  will  commence,  which  will  continue  for 
a  very  short  time,  till  the  bodies  have  acquired  a 
common  velocity.     The  two  bodies  will  then  cease  to 


DYNAMICS.  165 

act  upon  each  other,  and  will  move  with  the  common 
velocity  as  one  body. 

Let  the  masses  of  A  and  B  be  denoted  by  m  and 
m\  and  their  velocities  before  meeting  by  v  and  v' 
respectively,  and  let  the  common  velocity  which  they 
have  after  meeting  be  denoted  by  u :  then  the  velo- 
city lost  by  A  will  be  v  —  u,  and  the  velocity  gained 
by  B  will  be  u  —  v' ,  and  the  quantities  of  motion  lost 
and  gained  will  be  m  (v  —  u)  and  m'  (u  —  v')  respec- 
tively. Consequently,  since  by  the  third  law  of  mo- 
tion the  force  or  quantity  of  motion  lost  by  A  must 
be  equal  to  that  gained  by  B,  we  shall  have 

7n  {v  —  u)  =  ?7i'  {u  —  v')  ; 

and  hence 

7nv  -f-  m'v'  r  T  » 

u= I      ,      [a]* 

m-\-m'  -^ 

The  equation  for  the  case  in  which  the  two  bodies 
move  in  opposite  directions,  may  be  immediately 
deduced  from  equation  [a]  by  making  v'  negative. 
We  shall  thus  have,  for  this  case, 


,      ,     [bj 

54.  Application  of  equations  [a]  and  [b]  to  particular 
cases. 

1°.  Let  one  of  the  bodies,  as  B,  be  supposed  at  rest  * 

*  This  result  may  also  be  found  by  the  principle  of  D'Alera- 
bert ;  thus  mv-{-m'v'  is  the  impressed,  and  {?n -\- ?)i')  u  is  the 
effective  quantity  of  motion  :    and  hence 

mv  -f-  wi'y'  —  (m  -|-  ni')  u  =  0, 

and  u  = \ —  ,    , 

m  -\-  rrv 


166  MECHANICS. 

then  v'  =0,  and  we  shall  have 

771 V 

m  -\-  m' 

In  this  case,  we  perceive  that  as  m'  increases,  v 
and  m  being  supposed  to  remain  the  same,  u  dimin- 
ishes; and  that  when  m'  is  so  great  that  it  may  be 
considered  infinite  with  respect  to  ?n,  u  is  zero. 

2^.  Let  the  bodies  be  supposed  equal  in  mass ;  then 
m  =  m'\  and  according  as  they  move  in  the  same 
or  in  opposite  directions,  we  shall  have 

u  =  J-(r  -j-  I-'),     or  2^  =  J(y  —  c'). 

3°.  Let  the  bodies  be  supposed  to  be  equal  in  mass, 
and  one  of  them,  as  B,  to  be  at  rest :  then  m  =  m' 
and  i;'=0,  and  we  shall  have 

u  =  Iv. 

65.  Elastic  bodies. 

When  a  perfectly  elastic  body  of  s]3herical  form 
impinges  at  right  angles  upon  a  fixed  plane,  its  velo- 
city is  gradually  diminished,  and,  when  the  compres- 
sion reaches  its  limit,  is  reduced  to  zero;  but  the 
instant  the  compression  is  completed,  the  body,  in 
virtue  of  its  elasticity,  begins  to  resume  its  primitive 
figure,  and,  in  the  operation,  acquires  a  velocity  in 
the  opposite  direction,  exactly  equal  to  that  which 
was  lost.  Let  us  apply  this  result  to  the  collision  of 
two  perfectly  elastic  spherical  bodies  A  and  B  [Fig. 
29],  supposed  to  be  moving  from  left  to  right  on  the 
line  joining  their  centres ;  and  let  the  same  notation 
be  employed  as  in  the  preceding  case.  Art.  53.  In 
estimating  the  effect  of  the  collision,  it  is  evident  that 


DYNAMICS.  167 

until  the  elasticity  begins  to  act,  we  may  consider  the 
bodies  as  inelastic.  Consequently  at  the  instant  of 
greatest  compression,  the  velocity  u  common  to  each 
body  will  be  given  by  the  equation 

^^mv-^m'v'      ^^^ 

and  the  velocities  lost  by  A  and  gained  by  B  during 
the  compression,  will  be 

V  —  u  and  u  —  r'. 

But  in  the  process  of  resuming  its  primitive  figure, 
each  body  may  obviously  be  regarded  as  acting  upon 
a  fixed  plane,  supposed  to  pass  through  the  point  of 
meeting  of  the  two  bodies,  at  right  angles  to  the  line 
on  which  they  are  moving.  Hence  during  this  pro- 
cess A  will  lose,  and  B  will  gain,  the  additional  velo- 
cities 

V  —  u  and  u  —  v' 

respectively,  and  the  total  velocities  lost  and  gained 
by  the  two  bodies  will  be 

2(2?  —  II)  and  1{ii  —  v') 

respectively.  If  then  we  denote  the  velocity  of  A 
after  collision  by  i;  ^  and  the  velocity  of  B  by  v  ^^,  we 
shall  have 

V,  =  v  —  2{v  —  u), 

and  i,^^^y/_j_  2(2^  —  1;'), 

or  V,,  =  2u  —v' ; 

and  substituting  in  these  equations  the   value  of  u 

given  by  equation  [c],  we  shall  get 


168  MECHANICS. 

2{mv4-?n'v') 
m  -\-  m 

and  ^    _2{7nv  +  m'v')_^^, 

"  m  -\-  m'  ' 

or,  reducing 

•  _  v{m  —  771')  +  2m'v'  .,-, 

and  ^  -^'K  — ^)  +  ^^^  [ei 

To  adapt  these  equations  to  the  case  in  which  the 
bodies  move  in  opposite  directions,  we  have  only  to 
suppose  the  velocity  v'  negative.     We  shall  thus  get 

V  (m,  —  m')  —  2  ???/  v'  r  r-  n 

m^m' '    tf] 

_  v'{m  —  m')  +  2mv  .  -, 

"""  ^rir+^'         l^SJ 

56.  Application  of  equations  [d]  and  [e]. 
1°.  Letw  =  w':  then 

!•/  =  v',  and  V,,  =  v. 
Thus  when  the  masses  are  equal,  the  collision  will  cause 
the  bodies  to  exchange  velocities. 
2°.  \jQt  m^^m'  and?;'  =  0:  then 

V,  =  0,  and  v,,  =  v. 

That  is,  the  body  A  will  be  brought  to  rest,  and  B  will 
acquire  its  entire  velocity ;  a  result  immediately  dedu- 
cible  from  the  preceding  case. 

From  this  it  is  evident,  that  in  the  case  of  a  series 

of  elastic  balls.  A,  B,  C, P,  Q,  R,  of  equal  mass, 

in  contact  with  each  other,  and  having  their  centres 
on  the  same  straight  line :  if  the  first  ball  (A)  be 
made  to  impinge  with  any  velocity  directly  upon  the 


DYNAMICS.  169 

second  (B),  the  only  visible  effect  will  be  to  cause  the 
last  of  the  series  (R)  to  move  in  the  same  direction  with 
an  equal  velocity,  all  the  intermediate  halls  remaining  at 
rest. 

3°.  When  i;'  =  0,  but  m  is  not  equal  to  m' ,  the  direc- 
tion of  the  motion  of  the  impinging  ball,  after  colli- 
sion, depends  upon  the  relative  values  of  m  and  m\ 
Thus  when  m~>  m\  v ,  and  v ,  ^  are  both  positive,  and 
both  halls  will  move  after  collision  in  the  original  direc- 
tion;  but  when  m<Cm',  v^^  is  positive,  but  v,  nega- 
tive ;  and  the  impinged  hall  only  will  move  in  the  origi- 
nal direction,  while  the  impinging  hall  will  rehound, 

57.  Application  of  equations  [f  ]  and  [g]. 

1*^.  hetm  =  m':  then 

V/  =  —  u',  and  V/,  =  v. 

Thus,  in  this  case,  the  hodies  will  exchange  hoth  velo- 
cities and  directions. 
2^.  Let  v'  =v:  then 


If^,   and^.=^^(-^^-^^^> 


m-{-in' 

and  we  perceive  that  when  m  =  ^m\  we  have 

V,  =  0,  and  v,,  ~  2v. 

58.  Relative  velocity  before  and  after  collision. 
From  the  equations  v  ,  =  2u  — v,  and  v  ,,  =  2u  —  v' , 
we  get  by  subtraction 

V,  —  v„  =  —  {v  —  v')  ; 
from  which  it  appears  that  the  relative  velocities  of  the 
two  hodies  before  and  after  collision  are  equal  to  each 
other. 

22 


170  MECHANICS. 

59.  Imperfectly  elastic  bodies. 

In  the  collision  of  an  imperfectly  elastic  ball  with 
a  fixed  plane,  the  ratio  of  the  force  or  quantity  of 
motion  destroyed  by  the  compression  of  the  ball  (or 
the  force  of  compression),  to  the  force  generated  by  its 
restoration  to  its  primitive  figure  (or  the  force  of  resti- 
tution), is  called  the  modulus  of  elasticity  of  the  sub- 
stance of  the  ball.  If,  then,  in  any  case,  we  denote 
the  modulus  by  e,  the  mass  of  the  ball  by  m,  and  the 
velocities  corresponding  to  the  forces  of  compression 
and  restitution  by  c  and  r  respectively,  we  shall  have 

_  mr  _  r 
mc         c 

and  hence  r  =  ec [li] 

The  value  of  e  will  obviously  always  lie  between  zero 
and  unity  ;  the  former  limit  answering  to  the  case  of 
an  inelastic  body,  the  latter  to  that  of  a  body  per- 
fectly elastic. 

To  determine  the  velocities  after  collision,  in  the 
case  of  two  imperfectly  elastic  balls  moving  on  the 
line  joining  their  centres,  we  proceed  in  the  same 
manner  as  in  Art.  55.  Thus,  employing  the  same 
notation  as  in  that  article,  the  velocities  lost  by  A 
and  gained  by  B  during  the  compression,  being 

V  —  u  and  u  —  v', 

the  velocities  lost  and  gained  by  the  balls  in  the  pro- 
cess of  recovering  their  figure,  will  be  [equation  h], 

e{v  —  2i)  and  e{u  —  v'), 

and  the  total  velocities  lost  and  gained  will  be 

{v  —  ii)  -\-  e{v  —  ?/)  and  {21  —  v')  -f  ^(^^  —  "?'')• 


DYNAMICS.  171 

Hence  we  shall  have 

v,  =  v  —  {v  —  u)  —  e{v  —  u) 
=  (1  -|-e)  li  — ev 

=  (1  +  e) ^ — J—  —  ev,   [i] 

and  V,,  =  v'  +  {u  —  v')+e  {u—  V') 

=  {1  -\-  e)  u  —  ev' 

=  (1  +  e) \—. ev' [k] 

If  we  sup230se  the  balls  to  have  equal  masses,  and 
A  to  impinge  upon  B  at  rest,  we  have 

m=m'  and  t)'  =  0  ; 

and  hence  ^^^=  (i_|_e)_^^; 

tit 

from  which  we  get 

e  =  — ^'  —  1 Iml 

To  determine  in  a  given  case  the  modulus  of  elas- 
ticity, we  suspend  at  the  points  C  and  C  in  the  man- 
ner represented  in  Fig.  30,  two  equal  spherical  balls 
A  and  B  of  the  substance  in  question ;  and  withdraw- 
ing the  one  (A)  from  its  position  of  equilibrium,  suffer 
it  to  descend  and  impinge  upon  the  other  (B)  at  rest. 
The  arcs  described  by  the  respective  balls  being 
MNG,  M'  N'  G' ,  the  velocities  v  and  v ,  ^  with  which  A 
impinges  upon  B,  and  which  B  acquires  from  the  im- 
pulse, will  be  given  [Art.  17]  by  the  equations 


i;^  =  2gXPO=2^X 


vJ=--2gX  V'Cy'=^2g  X 


FG  ' 

M'G^^ 
F'G'  ' 


172  MECHANICS. 

Hence,  since  FG  =  F'G',  we  have 

V         \  PG"        MG  ' 

and  substituting  this  value  of  ~  in  equation  [m],  we 
get 

-"^^^-MG""^- 

The  arcs  MNG,  M'N'G'  being  known,  their  versed 
sines  PG,  P'G',  or  their  chords  MG,  M'G',  can  be 
easily  calculated,  and  hence  the  value  of  e  determined. 

60.  Loss  of  living  force  in  the  collision  of  inelastic 
bodies. 

The  product  of  the  mass  of  a  material  point  or  of 
a  body,  all  the  points  of  which  have  the  same  velo- 
city, by  the  square  of  its  velocity,  is  called  the  living 
force,  or  vis  viva,  of  the  point  or  body. 

Thus,  employing  the  same  notation  as  in  article 
53,  the  sum  of  the  living  forces  of  the  inelastic 
spheres  A  and  B,  moving  in  the  manner  supposed  in 
that  article,  is,  before  collision,  mv^  +  m'v'^',  and  after 
collision,  (m-\-m')u~.  Let  the  latter  of  these  expres- 
sions be  subtracted  from  the  former,  and  let  their 
difference  be  denoted  by  d ;  then  we  shall  have 

d  —-  mv^  -\-  m'v'  "^  —  (^'^  "j-  'in')u^. 

But  multiplying  equation  [a],  (Art.  53),  by  {m  +  m'  )u, 
we  have 

{m  -\-  m')u'^  =  muv  -j-  m'uv'^ 

and  hence 


DYNAMICS.                                              173 
0  =  miiv  -\-  m'uD'  —  {m  -f-  m')!^^ ;   [n] 

and  adding  these  two  equations,  we  get 

{m  -\-  ??i')u'^  =  2??iuv  -\-  2?}i'iiv'  —  (/?i  -\-  7n')u^. 
Consequently  we  have 

d  =  ?nv^  -\-  m'v'  "^  —  Iniuv  —  2m'uv'  -j-  miL^  -j-  m'u^ 
=  m{v  —  uf  -\-vi'{u  —  v'y. 

The  second  member  of  this  equation  being  positive, 
it  follows  that  in  the  case  of  the  two  inelastic  spheres 
A  and  B,  the  sum  of  the  living  forces  after  collision  is 
less  than  their  sum  before  collision.  It  is  also  evident 
from  the  form  of  the  second  member,  that  the  differ- 
ence of  the  two  sums,  or  the  loss  of  living  force  by 
the  collision,  is  equal  to  the  sum  of  the  living  forces 
due  to  the  velocities  {v  —  u)  and  {u  —  v')  w^hicli  are 
lost  and  gained  by  the  bodies  respectively.  This  is 
a  particular  case  of  the  general  principle,  that  the 
sudden  changes  of  velocity  which  occur  in  the  colli- 
sion of  inelastic  bodies,  whatever  their  number  and 
forms,  and  whether  the  collision  takes  place  among 
themselves  or  with  fixed  obstacles,  are  always  attended 
with  a  loss  of  living  force. 

61.  Conservation  of  living  force  in  the  collision  of 
perfectly  elastic  bodies. 

If,  as  in  article  55,  we  suppose  the  two  spheres  A 
and  B  to  be  perfectly  elastic,  employing  the  same 
notation  as  in  that  article,  we  shall  have 

V,  =  2u  —  i;  and  v,i  =  1u  —  v'  \ 

and  hence 

mv^  =  m{2u  —  vY  and  m'v,l^  =  vi'{2u  —  v'f. 


174  MECHANICS. 

Consequently  we  shall  have 

mv;  +  m'v,r  =  wz(4z^^  —  4:7iv  +  v')  +  7?i'{hi'  —  4^^b•'  +  v") 

=  4:{??iu''  -\- 771' it^  —  muv  — m'uv')  -\-  vi\f  +  ^^^'^' ^  J 

and  since  [Equa.  n]  the  expression  within  the  paren- 
thesis is  equal  to  zero,  we  shall  get 

mv,"^  -f-  m'Cii^  =  m\f  -\-  m'v' ^ 

Thus  in  the  case  of  the  two  perfectly  elastic  spheres 
A  and  B,  the  sum  of  the  living  forces  before  and  after 
collision  is  the  same.  This  is  a  particular  case  of  the 
general  principle,  that  in  the  collision  of  perfectly  elas- 
tic bodies,  there  is  no  loss  of  living  force. 

62.  Conservation  of  the  ?notion  of  the  centre  of  gravity 
in  the  collision  of  bodies. 

Let  the  distances  of  the  centres  of  the  two  spheres 
A  and  B  from  any  assumed  point  of  the  line  on  which 
they  are  moving,  at  any  instant  before  collision,  be 
denoted  by  x  and  y;  and  let  the  distance  of  their 
common  centre  of  gravity  from  the  same  point,  at  the 
same  instant,  be  denoted  by  z:  then,  retaining  the 
previous  notation,  we  shall  have  (Art.  45,  Statics) 

(??z  -|-  }fi')z  =  mx  -\-  ni'y. 

Again,  let  the  corresponding  distances  from  the  same 
point,  for  the  instant  immediately  following,  be  de- 
noted by  x' ,  y'  and  z'  :  we  shall  also  have 

[7)1  -f-  7n')z'  =  7ux'  ~\-  m'y'. 

Hence,  subtracting  the  first  equation  from  the  second, 
we  shall  have 

{???  -j-  ???')  (z'  —  z)  —  7n{x'  —  ,r)  -f-  ???'(?/'  —  ?/)  ; 


DYNAMICS.  175 

and  denoting  the  infinitely  small  interval  between  the 
two  instants  by  t' ,  and  dividing  each  member  of  this 
equation  by  it,  we  shall  get 

('»+'"')^-"'C^-)+"''(-7-^)- 

If  we  suppose  the  second  instant  to  be  that  which 
immediately  precedes  the  collision,  ^~^~  and  ^~~  ^  will 
respectively  express  the  velocities  which  we  have 
denoted  by  v  and  v'  :  Also  — p^  will  express  the  velo- 
city, which  we  will  denote  by  V,  of  the  common 
centre  of  gravity  of  the  two  bodies.  Hence  we  shall 
have 

{7n  -j-  m')  V=  mv  -f-  m'v', 
and  y_^mv  +  7n'v' 

If  we  denote  the  velocities  of  A  and  B  immediately 

after  collision,  by  w  and  w' ,  and  the  velocity  of  their 

common  centre  of  gravity  at  the  same  instant  by  F', 

we  shall  find,  by  an  operation  similar  to  the  above, 

y,  _  mw  -j-  m'w' 
m  -{-  m! 

We  will  apply  these  results  to  the  two  extreme 
cases. 

1°.  When  the  two  bodies  are  inelastic,  they  have, 
after  collision,  a  common  velocity  u.  Hence,  for  this 
case,  we  have 

W—W'  =  21, 

and  Y>^i^2L+^h  =  u. 

m-\-  m' 
But  Art.  [53]  ^^mv-\-m'v'  , 


m  -\-  m! 


176  MECHANICS. 

hence  [Equa.  o]  V=u, 

and  consequently         V=V'. 

2°.  AYlien  the  bodies  are  perfectly  elastic,  we  have 

w  =  V,  =  2u  —  v^  and  w'  =v,,  =  2u  —  u' ; 

and  hence 

y,  _  7n{2u  —  v)  -f-  m'{2u  —  v') 
m  -\-  m' 

m  -\-  m 
■=  2u  —  u  =  u. 
But  V=  u ; 

consequently  F=  v. 

Thus  whether  the  two  bodies  are  inelastic  or  per- 
fectly elactic,  the  velocity  of  their  common  centre  of 
gravity,  immediately  before  and  after  their  meeting, 
is  the  same ;  the  collision,  though  changing  the  velo- 
city of  each  body,  producing  no  alteration  in  the 
velocity  of  that  point.  This  is  a  particular  case  of 
the  general  principle,  called  "  the  principle  of  the  con- 
servation of  the  motion  of  the  centre  of  gravity,"  that 
the  mutual  action  of  the  bodies  of  any  system  does  not 
alter  the  motion  of  the  centre  of  gravity  of  the  system. 

63.   Oblique  collision. 

We  shall  consider  only  the  simple  case  of  the  col- 
lision of  a  spherical  ball  Avith  an  immovable  plane. 
Let  AB  [Fig.  31]  represent  the  plane,  and  CI  the  line 
described  by  the  centre  of  the  ball.  On  CI  produced, 
let  IE  be  taken  to  represent  the  velocity  of  the  ball ; 
and  let  it  be  resolved  into  the  components  IF  and 
IM,  the  one  in  the  plane  AB,  the  other  perpendicular 


DYNAMICS.  177 

to  it.     Denoting  the  velocity  of  the  ball  by  v^  and 
CIL  the  angle  of  incidence  by  a,  we  shall  have 
IF  =  V  sin  a,  and  IM  =  v  cos  a. 

1°.  If  the  ball  is  inelastic,  there  will  be  no  compo- 
nent perpendicular  to  the  plane  after  collision,  and 
the  ball  will  move  in  the  plane  with  the  velocity 
V  sin  a. 

2°.  If  the  ball  is  perfectly  elastic,  it  will  tend  to 
rebound  in  the  direction  IL  with  the  velocity  IN, 
equal  to  IM  or  v  cos  a  ;  and  in  virtue  of  the  two  velo- 
cities IF  and  IN,  will  move  in  the  direction  IG  with 
a  velocity  ID  equal  to  IE ;  the  angle  of  reflection  GIL 
being  equal  to  the  angle  of  incidence. 

3°.  If  the  ball  is  imperfectly  elastic,  it  will  tend  to 
rebound  with  a  velocity  IN'  less  than  IN,  and  such, 
that  if  e  denote  the  modulus  of  elasticity,  we  shall 
have 

_IN' 

""""IN' 

and  hence  IN'  =  e  x  IN. 

In  this  case,  the  ball  will  move,  after  collision,  in  the 
direction  IG' ,  and  with  a  velocity  ID'  which  will  be 
given  by  the  equation 

ID'  =  a/v"^  sin^  a  -{-  t'^v^  cos'^  a ; 

and  if  we  denote  the  angle  of  reflection  LIG'  by  a\ 
this  angle  will  be  given  by  the  equation 

,           ,      \)  sin  a       tanF  a 
tano-  a'  = =  — ^ — . 


23 


178  MECHANICS. 

GRAVITATION. 

64.  It  has  been  found  by  observation,  that  the 
planets,  in  their  revohitions  about  the  sun,  observe 
the  following  laws : 

1st.  The  areas  described  hy  the  radius  vector  of  a 
planet  (or  the  line  drawn  from  the  centre  of  the  sun 
to  the  centre  of  the  planet),  are  as  the  times  employed 
in  describing  them. 

2d.  The  orbit  of  a  planet  is  an  ellipse  of  small  eccen- 
tricity, having  the  centre  of  the  sun  at  one  of  its  foci. 

3d.  The  squares  of  the  periodic  times  of  any  two  planets 
(or  the  times  of  the  complete  revolution  of  each  about 
the  sun),  are  to  each  other  as  the  cubes  of  their  mean 
distances  from  the  sun  (or  the  semi-transverse  axes  of 
their  orbits.) 

The  satellites,  in  their  revolutions  about  their  pri- 
maries, are  found  to  obey  the  same  laws. 

These  laws  are  called,  from  the  name  of  their  dis- 
coverer, Kepler^s  laws. 

According  to  the  first  law  of  motion,  the  tendency 
of  a  planet,  at  each  point  of  its  orbit,  is  to  move  in 
the  straight  line  which  is  tangent  to  its  orbit  at  that 
point:  therefore,  since  its  motion  is  curvilinear,  it 
must  constantly  be  acted  upon  by  an  accelerating 
force.  The  object  of  this  section  is  very  briefly  to 
investigate  the  laws  which  regulate  this  force.  The 
laws  of  Kepler  refer  to  the  motions  of  the  centres  of 
gravity  of  the  planets;  and  in  all  that  follows,  we 
shall  consider  the  masses  of  the  planets  as  reduced  to 
these  points. 


DYNAMICS.  179 

1°.   The  direction  of  the  force. 

Since  the  areas  described  by  the  radius  vector  of 
a  planet  are  proportional  to  the  times  employed  in 
describing  them,  the  force  which  deflects  the  planet 
from  the  straight  line  in  which  it  tends  to  move, 
must  (Art.  22)  coincide  in  direction  with  the  radius 
vector,  and  constantly  solicit  the  body  towards  the  centre 
of  the  sun. 

2°.  Law  according  to  which  the  intensity  of  the  force 
varies. 

Let  AMA'  [Fig.  32]  represent  the  elliptical  orbit 
of  a  planet,  and  F  the  focus  occupied  by  the  centre 
of  the  sun.  Let  MM"  be  an  arc  described  by  the 
planet  in  the  indefinitely  short  time  /'  ;  and  to  its 
extremities  M  and  M' ' ,  draw  the  radii  vectores  FM 
and  FM' ' .  Draw  MY  tangent  to  the  curve  at  M ; 
and  from  M  and  F,  draw  MC  and  FY  perpendicular 
to  MY.  Draw  also  M"  N,  M"  K  perpendicular  to  MC 
and  MF  respectively;  and  take  MC  equal  to  the 
radius  of  curvature  at  the  point  M. 

The  arc  MM"  being  infinitely  small,  the  direction 
and  intensity  of  the  accelerating  force,  during  the 
time  f ,  may  be  supposed  to  remain  the  same.  Dur- 
ing this  time,  then,  the  accelerating  force  acting- 
alone  would  cause  the  point  to  describe  the  line  MI. 
Hence  denoting  the  intensity  of  the  force  by  F,  we 
have  [Art.  3], 

and  taking  V  for  the  unit  of  time,  we  get 

F  =  2MI [a| 


180  MECHANICS. 

To  find  an  expression  for  MI,  we  get  from  the  sim- 
ilar triangles  MNI,  M"KI, 

M"K     :     M"I     :  :     MN     :     MI; 

and  hence  mi  =  MN  x  4^ [1^1 

M'K 

But  since  MM' '  may  be  considered  the  arc  of  a  circle 
of  which  MC  is  the  radius,  we  have 

M"N^  =  (2MC'  —  MN)MN  ; 
or,  neglecting  MN  m  comparison  with  2MC', 
M"N^  =  2MC'  X  MN  ; 

and  hence  mN  =  M!^ 

2MC' 

2MC' '-^-' 

Moreover,  we  have 

{p  denoting  the  parameter  of  the  ellipse) ;  or,  observ- 
ing that  the  similar  triangles  M' '  KI,  F YM  give 
FM     :     FY     :  :     M"!     :     M"K, 

or  FM  _  U"l 

FY      M'K' 

we  have  MC  =  ^  x  /MILV : 

2^  W'k) 

hence  substituting  this  value  of  the  radius  of  curva- 
ture in  equation  [c],  we  get 

*  NI  may  be  neglected  in  comparison  with  M"I,  since  it  is  a  quantity 
of  the  same  order  as  MN ;  that  is,  an  infinitely  small  quantity  of  the 
second  order. 

t  Jackson's  Conic  Sections,  Chap.  IV,  Prop.  I,  Cor.  2. 


i 


DYNAMICS.  181 

^i^^=ix-K- [^J 

Lastly,  substituting  this  value  of  MN  in  equation  [b], 
we  have 

MI  =  ^  X  M"K^ [e] 

P 

If  now  we  denote  the  area  of  the  elliptical  sector 
FMM' '  by  s,  we  have  (considering  MM' '  as  a  straight 
line), 

FM  X  M"K 

and  hence  M"K'  =  ^—  • 

and  substituting  this  value  of  M' '  K^  in  equation  [e], 
we  get 

p       FM^ 

hence  [Equa.  a]  we  have 

^  =  fx#^ m 

Again,  considering  the  action  of  the  accelerating 
force  at  some  other  point  of  the  ellipse,  as  M' ,  and  for 
an  equal  interval  of  time  (f),  we  have,  denoting  the 
intensity  of  the  force  by  i^ ' , 

p  ^  FM" 

consequently  we  have 

F     :     F'     :  :     FW     :     YW. 

Thus  the  force  which  retains  the  planet  in  its  orbit, 
varies  in  the  inverse  ratio  of  the  square  of  the  distance 
of  the  planet  from  the  centre  of  the  sun. 


182  MECHANICS. 

65.   Variation  of  the  force  from  one  planet  to  another. 

Let  T  and  T'  denote  the  periodic  times  of  two 
planets ;  s  and  s' ,  the  areas  described  by  their  radii 
vectores  in  the  time  t' ;  and  a  and  b,  a'  and  b' ,  the 
semi-axes  of  their  orbits :  then  the  area  described  by 
the  radius  vector  of  the  first  phanet,  in  the  time  T, 
will  be  Ts ;  but  this  area,  being  the  entire  surface  of 
the  ellipse,  will  also  be  expressed  by  -ab  :  hence  we 
shall  have 

Ts  =  TTah 
and  rp^Trnb^ 

s 

We  shall  also  have 

consequently,  according  to  Kepler's  third  law,   we 
shall  have 

^^     '-     -^     '-''      ^       '-     - 

or,  multiplying  the  antecedents  by  ^  and  the  conse- 
quents by  ^,, 

a  a' 

or,  denoting  the  parameters  of  the  ellipses  by  jd  and  p' , 

,2    .     „,.    .  .     p    .    p' 

.s      .     s       '   ■       2      '      2  • 

But  denoting  the  accelerating  forces  which  act 
upon  the  two  planets  at  the  points  M  and  M ,  of  their 
orbits,  by  F  and  F ,,  we  have  (Equa.  f,  Art.  64) 

p    •    F     •  •    -—-V  J—    •    JL  V  -1 

'    '  '     p  '^  YW   '    p'  ^  fm;^' 


DYNAMICS.  183 

Combining  then  this  proportion  with  that  immedi- 
ately preceding,  we  get 

From  this  we  infer  that  the  force  which  solicits  the 
planets  towards  the  centre  of  the  sun,  varies  from  one 
planet  to  another  according  to  the  same  law  as  that 
lohich  governs  in  different  positions  of  the  same  planet ; 
the  variation  depending  only  upon  the  distance,  and 
not  upon  any  peculiarities  that  may  exist  in  the  con- 
stitution of  the  planets  themselves. 

66.  Since  the  satellites,  in  their  revolutions  about 
their  primaries,  also  conform  to  Kepler's  laws,  the 
force  which  retains  them  in  their  orbits  must  solicit 
them  towards  the  centres  of  their  primaries,  accord- 
ing to  the  same  law  as  that  which  obtains  in  the  case 
of  the  sun  and  planets. 

67.  From  the  foregoing  results,  it  appears  that  the 
jDhenomena  take  place  as  if  the  matter  of  the  bodies 
composing  the  solar  system  were  endowed  with  a 
peculiar  property,  in  virtue  of  which  these  bodies 
exert  a  mutual  attraction  according  to  the  above  law. 

With  respect  to  the  relation  between  the  attrac- 
tion exerted  by  a  body,  and  the  mass  of  the  body, 
the  most  obvious  supposition,  viz :  that  the  force  is 
directly  proportional  to  the  mass,  is  found  to  agree 
with  the  results  of  observation  and  calculation. 

Lastly,  it  can  be  shown,  by  methods  not  adapted 
to  the  present  work,  that  if  bodies,  considered  in  the 
aggregate,  attract  each  other  in  the  inverse  ratio  of 


184  MECHANICS. 

the  squares  of  their  mutual  distances,  then  their  ulti- 
mate particles  must  attract  each  other  according  to 
the  same  law.  We  thus  arrive  at  the  general  law, 
that  the  particles  of  all  bodies  attract  each  other  in  the 
direct  ratio  of  their  masses,  and  the  inverse  ratio  of  the 
squares  of  their  distances.  This  law  is  called  the  law 
or  theory  of  universal  gravitation.  Its  full  development 
in  its  application  to  the  bodies  of  the  solar  system, 
constitutes  the  science  of  physical  astronomy.  The 
agreement  of  the  deductions  of  this  science  with  the 
results  of  observation,  is  such  as  perfectly  to  establish 
the  truth  of  the  law. 


PART  THIRD. 


HYDROSTATICS, 


1.  The  conception  of  a  fluid,  in  mechanics,  is  that  of 
a'  substance  in  which  the  ultimate  particles  can  be 
moved  amongst  each  other  by  the  application  of 
any  force,  however  small.^  Fluids  are  divided  into 
two  classes : 

I.  Compressible  or  aeriform  fluids  ; 

II.  Incompressible  fluids,  or  liquids.f 
Atmospheric  air  is  usually  taken  as  the  type  of 

the  first  class ;  water,  as  that  of  the  second. 

The  basis  of  the  mechanics  of  fluids  is  a  principle 
derived  from  experiment.  If  a  vessel  having  two 
or  more  apertures  of  equal  areas,  closed  by  pistons, 
be  filled  with  water,  and  such  forces  be  applied  to 

^^  Fluidity,  as  thus  defined,  must  be  regarded  as  a  limiting  state, 
which  fluids,  as  they  actually  exist,  approach  more  or  less  nearly. 
Thus  water,  alcohol,  and  the  ethers,  may  be  regarded  as  sensibly  in 
this  state;  while  molasses,  and  the  more  adhesive  oils,  approach  it 
much  less  nearly. 

t  Liquids,  as  they  actually  exist,  are  not  rigorously  incompressible ; 
but  the  compression  of  which  they  are  susceptible  is  so  very  small, 
that,  in  establishing  the  principles  of  hydrostatics,  it  is  not  taken  into 
account. 

24 


186  MECHANICS. 

the  pistons,  supposed  to  move  freely  at  right  angles 
to  the  surface  of  the  water,  as  will  produce  an  equi- 
librium, then  if  the  pistons  be  pressed  inwards  per- 
pendicularly by  the  forces  P,  P',  P",  etc.,  the  equi- 
librium will  continue  only  when  P  —  P'  —  P'\  etc. 

Thus,  let  ABCDE  [Fig.  1]  be  a  vertical  section  of 
such  a  vessel,  in  which  the  equal  apertures  F,  G  and 
H  are  closed  by  pistons  whose  w^eights  are  so  adjusted 
as  to  produce  an  equilibrium ;  then  it  will  be  found, 
that  if  we  apply  to  one  of  the  pistons  a  force  P,  to 
maintain  the  equilibrium,  w^e  must  apply  to  the 
others  severally  the  forces  P',  P",  each  equal  to  P. 
In  whatever  manner  the  experiment  be  varied,  the 
same  result  wall  always  be  obtained.  It  thus  appears 
that  a  pressure  applied  to  the  surface  of  a  fluid  at  rest, 
is  transmitted  unchanged  in  intensity  in  all  directions 
throughout  the  fluid ;  so  that  equal  portions  of  the  surface 
of  the  containing  vessel,  supposed  plane,  are  subjected 
thereby  to  equal  pressures,  and  unequal  portions  to  pressures 
proportional  to  their  areas. 

If  w^e  denote  by  P  the  pressure  upon  the  area  A, 
and  hjp  the  pressure  upon  the  area  w^hich  is  assumed 
as  the  unit  of  surface,  we  shall  have 
A    :    1    \:    P    .    p; 

and  hence  p  =    — -. 

When  the  surface  of  the  vessel  is  curved,  we 
regard  it  as  the  surface  of  a  polyhedron  of  an  infinite 
number  of  plane  faces,  each  infinitely  small;  and 
denoting  the  area  of  any  one  of  these  faces  by  a,  we 


i 


HYDROSTATICS.  187 

have,  for  the  expression  of  the  pressure  exerted 
upon  it,  the  product  pa ;  p  denoting,  as  above,  the 
pressure  upon  the  unit  of  surface. 

The  direction  of  the  pressure  upon  the  surface  of  the 
vessel,  at  any  point,  is  perpendicular  to  the  surface  at  that 
point ;  for  were  it  not,  it  could  be  resolved  into  two 
components,  the  one  perpendicular,  the  other  paral- 
lel to  the  surface,  the  former  of  which  would  be 
destroyed,  while  the  latter  would  produce  motion. 

2.  A  little  consideration  will  render  it  evident 
that  the  pressure  applied  to  the  surface  of  a  fluid  is 
transmitted  unchanged,  not  only  to  every  equal 
portion  of  the  surface  of  the  vessel  containing  it,  but 
also  to  every  equal  surface  or  equal  stratum  of  particles 
within  the  fluid.  This,  though  sufficiently  obvious  from 
w^hat  has  preceded,  may  be  rendered  clearer  to  some 
minds  by  the  following  method  of  establishing  it. 
Let  mn  [Fig.  2]  represent  a  stratum  of  particles  in 
the  interior  of  the  vessel  ABCD  :  then  the  pressure 
upon  the  stratum  will  not  be  changed  if  we  suppose 
the  surface  EF,  of  which  mn  forms  a  part,  to  become 
rigid,  or,  what  is  equivalent,  the  fluid  contained 
in  the  part  EDF  of  the  vessel  to  become  solid  ;  for 
an  equilibriun  which  exists  among  forces  acting 
upon  a  system  of  material  points,  is  not  disturbed 
by  supposing  any  number  of  these  points  to  become 
fixed ;  but  in  this  case,  the  surface  mn  is  in  precisely 
the  same  condition  as  if  it  formed  a  23ortion  of  the 
surface  of  the  vessel  :  w^hence  the  truth  of  the  pro- 
position is  evident. 


188  MECHANICS. 

3.  Pressure  due  to  the  gravity  of  the  fluid. 

In  addition  to  the  pressure  which  we  have  been 
considering,  the  particles  of  the  fluid,  and  hence  the 
surface  of  the  vessel,  are  subjected  to  a  pressure 
caused  by  the  gravity  of  the  fluid.  This  pressure, 
which  is  normal  to  the  surface  and  acts  from  within 
outwards,  obviously  varies  from  one  point  to  another 
of  the  fluid  or  surface.  Since  then  it  can  be  supposed 
constant  only  for  an  infinitely  small  space,  to  obtain 
a  measure  of  it,  we  conceive  a  plane  area,  assumed 
as  the  unit  of  surface,  to  be  pressed  at  all  its  points 
with  the  same  intensity  as  the  points  of  this  space  : 
then  denoting  the  pressure  upon  the  unit  by  p,  and 
the  extent  of  the  infinitely  small  surface  or  element 
by  a,  we  have,  for  the  expression  of  the  pressure 
upon  this  element,  the  product  pa.  The  coefficient 
p  is  called  the  pressure  referred  to  the  unit  of 
surface,  or  the  unit  of  pressure. 

The  attention  of  the  student  should  be  particu- 
larly directed  to  the  distinction  between  these  two 
kinds  of  pressures,  which  are  sustained  by  the  parts 
of  the  liquid  and  the  sides  of  the  vessel  containing 
it  :  the  first  caused  by  forces  applied  to  the  surface  of 
the  liquid,  and  the  same  at  every  point ;  the  second, 
due  to  the  weight  of  the  fluid,  and  varying  from  one 
point  to  another.  The  total  pressure  at  any  point 
is  obviously  the  sum  of  these  two  pressures. 

4.  Principle  of  virtual  velocities  observed. 

An  incompressible  fluid,  enclosed  in  a  vessel  fur- 
nished with  pistons,  may,  according  to  the  definition 


HYDROSTATICS.  189 

in  Art.  47,  Statics,  be  regarded  as  a  machine.  The 
principle  of  virtual  velocities  is  observed  in  the 
equilibrium  of  forces  applied  to  a  fluid  thus  treated, 
as  it  is  in  the  case  of  all  other  known  machines. 

Thus,  let  the  system  of  forces  P,  P ,  P  \  etc.,  be 
applied  to  an  incompressible  fluid  as  represented  in 
figure  3,  by  means  of  pistons  whose  areas,  supposed 
jDlane,  are  denoted  by  a,  a',  a",  etc.  These  forces 
being  in  equilibrium,  the  pressure  p,  referred  to  the 
unit  of  surface,  will  be  the  same  upon  every  portion 
of  the  interior  surface  of  the  vessel,  the  surface  of 
the  pistons  included ;  and  we  shall  have 
P    z=  ap,     P'  =  a'p,     F"  =  a"p,  etc. 

Now  if  we  suppose  the  pistons  to  be  displaced  through 
the  respective  spaces  h,  h',  h",  etc.,  the  sign  +  being 
prefixed  to  the  h  when  the  piston  is  moved  inwards, 
the  sign  —  in  the  contrary  case ;  we  shall  have, 
since  the  volume  of  liquid  is  invariable, 

ha-\-h'a'-\-h"a"-{-etQ.   =  0, 
or,  2^  ti  a  -{-  ph' a'  -{-  ph"  a"  -^  etc.  =  0; 

and  hence,  by  substitution, 

Ph-[-P'h'  +  P"h"  + etc.   =  0; 

which  is  the  principle  of  virtual  velocities,  in  a  more 
general  form  than  that  given  in  Statics,  Art.  69. 

If  there  are  only  two  pistons,  the  above  equation 
is  reduced  to 

Ph  +  P'h'  =  0; 

or,  since  h  and  h'  must  have  opposite  signs,  to 

X   Ph  —  P'h': 


190  MECHANICS. 

whence 

P    :    P    ::    h'    :    h, 

as  in  Art.  69. 

5.   Compressible  fluids. 

A  force,  applied  to  the  surface  of  a  compressible 
fluid,  is  propagated  throughout  the  fluid  in  all  di- 
rections, as  in  the  analogous  case  of  liquids.  Like 
liquids,  compressible  fluids,  and  the  surface  of  the 
vessels  containing  them,  are  also  subject  to  pres- 
sure from  the  action  of  gravity  upon  the  fluid ;  but 
they  difier  from  fluids  in  possessing  elasticity,  in 
virtue  of  which  their  particles  have  a  constant  ten- 
dency to  recede  from  each  other,  and  to  occupy  a 
greater  space.  Thus  a  mass  of  air,  or  any  other  gas 
contained  in  a  vessel  entirely  closed,  exe^rts  upon 
the  interior  surface  of  the  vessel  a  pressure  (in  addi- 
tion to  that  caused  by  gravity)  normal  to  the  surface, 
and  directed  from  within  outwards.  This  pressure  is 
called  the  elastic  force  of  the  gas,  and  is  the  same  at 
every  point ;  that  is,  the  pressure,  referred  to  the 
unit  of  surface,  is  the  same  throughout  the  mass. 
The  total  pressure,  on  account  of  the  action  of  gra- 
vity, varies  from  one  point  to  another  of  the  fluid. 

In  the  same  fluid,  at  a  constant  temperature,  the 
elasticit}^,  as  will  be  shown,  varies  directly  as  the 
density. 

An  elastic  fluid  obviously  cannot  be  in  equilibrium, 
unless  it  be  contained  in  a  vessel  entirely  closed,  or, 
like  the  atmosphere,  be  permitted  to  extend  indefi- 
nitely in  space  until  its  density  becomes  insensible. 


HYDROSTATICS.  191 

6.  The  equiUhrium  of  fluids,  regard  being  had  to  their 
gravity. 

The  surface  of  a  fluid  at  rest  is  horizontal,  that  is, 
perpendicular  to  the  direction  of  gravity.  For  if  the 
surface  were  oblique  to  the  direction  of  gravity,  we 
could  decompose  the  gravity  of  any  exterior  parti- 
cle of  the  fluid  into  two  components,  the  one  per- 
pendicular, the  other  tangent  to  the  surface,  the 
latter  of  which  would  cause  the  particle  to  glide  on 
the  surface  :  hence  an  equilibrium  cannot  exist,  un- 
less the  surface,  at  all  its  points,  be  at  right  angles 
to  the  direction  of  gravity,  or  horizontal.  Since  the 
directions  of  gravity  at  different  points  of  the  earth's 
surface  are  not  parallel  to  each  other,  the  surface  of 
a  liquid  at  rest  is  not  plane,  but  curved.  Surfaces 
of  small  extent  may,  however,  be  regarded  as  plane. 

7.  Pressure  exerted  by  a  liquid  upon  the  bottom,  sup- 
posed to  be  plane  and  horizontal,  of  the  vessel  containing  it. 

Let  ABDC  [Fig.  4]  represent  an  infinitely  thin 
vertical  slice,  of  uniform  thickness,  of  a  vessel  con- 
taining a  liquid,  open  at  top,  and  having  its  bottom 
plane  and  horizontal.  Conceive  the  Huid  whose  free 
surflice  is  ab  to  be  divided  into  indefinitely  thin 
horizontal  strata  by  the  planes  ab',  a"b",  etc. ;  and 
also  suppose  it  made  up  of  vertical  prisms,  mo,  pr, 
etc.,  whose  bases  are  equal  to  each  other  and  indefi- 
nitely small. 

If  we  denote  the  base  of  the  elementary  j^rism 
mm'  by  a,  and  its  height  by  k,  its  weight  will  be 
expressed  by  the  product  gkaD,  in  which  D  denotes 


192  MECHANICS. 

the  density  of  the  liquid  and  g  the  intensity  of  gra- 
vity. This  prism  will  exert,  in  the  fluid  below  the 
plane  a'b',  a  pressure  equal  to  gkaD  on  every  area 
equivalent  to  its  base  a,  and  will  thus  put  in  equi- 
librium all  the  other  elementary  prisms  pq,  etc.,  in 
the  upper  stratum  abb  a' ;  and  will  also  communi- 
cate to  the  bottom  CD  a  pressure  equal  to  gkaD, 
repeated  as  many  times  as  the  elementary  area  a  is 
contained  in  CD  ;  that  is,  a  pressure  equal  to  the 
weight  of  a  right  prism  of  the  liquid  whose  base  is 
CD  and  altitude  mm'.  The  same  being  true  of  all 
the  other  elementary  prisms  which  make  up  the 
prism  mo,  the  pressure  upon  CD,  and  hence  upon 
the  bottom  of  the  vessel,  will  be  equal  to  the  weight 
of  a  prism  of  the  liquid  whose  base  is  the  bottom, 
and  whose  altitude  is  mo.  Thus,  if  we  denote  the 
area  of  the  bottom  of  the  vessel  by  A,  its  distance 
below  the  free  surface  of  the  liquid  by  A,  and  the 
pressure  by  P,  we  shall  have 

P  z=i  gJiAD,     or  P  z=  pA, 

p  denoting  the  pressure  referred  to  the  unit  of  surface. 
This  result  is  obviously  true  of  every  horizontal 
plane  surface  immersed  in  the  liquid,  and  indeed  of 
every  horizontal  stratum  of  liquid ;  the  area  of  the 
plane  or  stratum  being  denoted  by  A,  and  its  dis- 
tance below  the  free  surface  by  h.  It  is  also  inde- 
pendent of  the  shape  of  the  vessel ;  so  that  while  the 
area  of  the  bottom  and  its  distance  below  the  free  surface 
are  constant,  the  pressure  upon  the  bottom  ivill  not  be 
changed,  however  the  shape  of  the  vessel  may  be  varied. 


HYDROSTATICS.  193 

8.  Now  to  the  mass  of  fluid  whose  pressure  ghAD 
upon  the  bottom  of  the  vessel  has  just  been  deter- 
mined, conceive  a  stratum  cdba  of  fluid  to  be  added, 
of  a  uniform  thickness  h'  and  a  density  D'  :  the  two 
fluids  will  be  in  equilibrium;  and  the  second  will 
exert  upon  the  upper  surface  ab  of  the  first,  equal 
pressures  at  all  its  points.  If  we  denote  this  surface 
by  A,  the  total  pressure  upon  it  will  be  expressed 
by  gh'A'D\  and  the  pressure  upon  the  unit  of  surface 
by  gh'D'.  This  pressure  will  be  transmitted,  by  means 
of  the  first  fluid,  to  the  bottom  of  the  vessel ;  and 
thus  a  new  pressure  will  be  exerted  upon  it,  ex- 
pressed by  gh'AB',  so  that  the  total  pressure  upon 
the  bottom  will  be  expressed  by 
gJiAD+gh'AD'. 

If  another  stratum  be  added,  the  equilibrium  will 
still  not  be  disturbed ;  and  if  we  denote  its  density 
and  thickness  by  D'  and  h"  respectively,  and  the 
upper  surface  cd  of  the  second  fluid  by  A",  the  pres- 
sure exerted  by  this  stratum  on  the  surface  A"  will 
be  expressed  by  gh"A"D",  and  the  pressure  upon 
the  unit  of  surface  by  gh!'D" .  This  pressure  will  be 
transmitted,  by  means  of  the  second  fluid,  to  the 
surface  A'  producing  upon  it  a  pressure  expressed 
by  gh"A'D";  and,  again,  by  means  of  the  first  fluid, 
it  will  be  transmitted  to  the  surface  A,  producing  a 
pressure  upon  it  equal  to  gh!'  AD' ,  so  that  the  entire 
pressure  of  the  three  fluids  upon  the  bottom  of  the 
vessel  will  be  expressed  by 

gliA  D  +  gh'AD'  +  g7i'>AD", 

or  {gJiD  +  gh'D'  +  gh"D")  A. 

25 


194  MECHANICS. 

The  same  reasoning  is  applicable,  whatever  the 
number  of  strata. 

It  thus  appears,  that  in  the  case  of  a  vessel  con- 
taining different  fluids  arranged  in  horizontal  strata, 
the  total  pressure  upon  the  bottom  of  the  vessel 
(supposed  to  be  horizontal)  depends  only  upon  the 
extent  of  the  bottom  and  the  thickness  and  density 
of  the  strata ;  being  equal  to  the  lueights  of  the  vertical 
columns  of  the  several  fluids  which  have  for  their  heights 
the  thicknesses  of  the  strata  respectively,  and  for  a  common 
base  the  bottom  of  the  vessel. 

When  the  fluids  are  arranged  as  we  have  sup- 
posed them  to  be  in  horizontal  strata,  that  is,  in  strata 
of  uniform  thickness,  they  are  evidently  in  equili- 
brium ;  and  it  can  be  easily  shown  that  an  equilibrium 
cannot  exist,  unless  they  are  thus  disposed.  It  is 
also  sufficiently  obvious  that  the  equilibrium  cannot 
be  stable,  unless  the  fluids  are  arranged  in  the  order 
of  their  densities,  the  most  dense  being  lowest. 

The  above  result  being  independent  of  the  thick- 
ness of  the  strata,  it  is  true  when  the  strata  are  infi- 
nitely thin,  that  is,  when  the  density  of  the  fluid 
varies  continuously.  It  is  also  true  when,  in  addi- 
tion to  this,  the  intensity  of  gravity  varies  from  one 
stratum  to  another.  Both  these  circumstances  meet 
in  a  column  of  air  extending  from  the  earth  to  the 
exterior  limit  of  the  atmosphere.  Hence  the  pres- 
sure of  such  a  column,  upon  a  plane  and  horizontal 
base,  is  equal  to  the  weight  of  the  i?ifi?iite  number  of 
successive  strata  of  which  it  may  be  supposed  to  consist. 


HYDROSTATICS.  195 

9.  When  several  vessels  are  so  connected  that  a 
fluid  can  flow  freely  from  one  into  another,  an  equi- 
librium will  evidently  exist  when  the  free  surfaces  of 
the  fluid  in  the  different  vessels  are  in  the  same  horizontal 
plane.  This  condition  is  also  necessary  to  an  equi- 
librium ;  for  if  the  free  surface,  GH  for  example,  of 
the  fluid  in  the  vessel  M  [Fig.  5],  when  produced, 
intersect  the  fluid  in  the  vessel  N,  not  in  EF  the 
plane  of  its  free  surface,  but  in  some  plane  DC  below 
it;  then  the  fluid  above  DC  will  exert  a  pressure 
upon  the  stratum  of  particles  in  DC,  and  hence  cause 
an  upward  pressure  in  the  stratum  GH  :  in  order, 
therefore,  to  an  equilibrium,  the  free  surfaces  must 
be  in  the  same  plane.  The  free  surfaces  then  being 
GH  and  EF,  let  two  new  fluids,  whose  densities  are 
D  and  D\  be  poured  into  the  vessels  M  and  N  re- 
spectively ;  and  let  the  heights  of  their  free  surfaces 
above  the  free  surfaces  of  the  first  fluid  be  denoted 
by  h  and  A',  and  the  areas  of  the  surfaces  GH  and  EF 
by  A  and  A  :  the  pressures  they  will  respectively 
exert  upon  the  surfaces  GH  and  EF  will  be  expressed 
by  ghAJD  and  gh'A'B',  and  their  pressures  upon  the 
unit  of  surface  by  ghD  and  gh'D' ;  and  hence,  in 
order  to  an  equilibrium,  we  must  have 

hD  =  h'D', 
or  h     :     h'     ::     D'     :     D; 

that  is,  the  heights  of  the  two  fluids  inversely  as  their 
densities. 

Whatever  the  number  of  connected  vessels,  it  may 


196  MECHANICS. 

in  like  manner  be  shown,  that  if  we  add  to  the  primi- 
tive fluid  in  each  any  number  of  different  fluids, 
in  order  to  an  equilibrium,  the  products  of  the  vertical 
heights  of  the  several  strata,  multiplied  by  their  respective 
densities,  must  he  the  same  for  each  vessel. 

When  an  equilibrium  exists,  it  will  evidently  not 
be  disturbed  if  all  communication  between  the  ves- 
sels be  prevented  by  the  interposition  of  fixed  and 
rigid  planes.  Hence,  to  determine  the  pressure  on 
the  bottom  and  sides  of  any  one  of  the  vessels,  we 
need  consider  only  the  fluid  which  that  vessel  con- 
tains. 

10.  Pressure  of  a  fluid  on  the  surface  of  the  vessel 
which  contains  it,  or  any  part  of  that  surface;  or  on  any 
immersed  surface,  plane  or  curved,  whatever  its  position. 

Let  mn  [Fig.  6]  be  an  infinitely  small  horizontal 
stratum  of  particles,  or  submerged  plane,  in  a  vessel 
containing  a  fluid.  If  the  area  of  the  plane  be  de- 
noted by  a,  the  pressure  it  sustains  from  the  gravity 
of  the  fluid  will  [Art.  3]  be  denoted  by  the  product 
ghaD ;  and  this  pressure  will,  according  to  the  fun- 
damental principle.  Art.  1,  be  communicated  to  all 
the  parts  of  the  fluid  below  the  plane  of  mn.  Let 
now  mn  revolve  about  anj^  one  of  its  points,  as  m, 
assumed  to  be  fixed,  and  take  any  position  mn'.  In 
this  new  position,  it  will  sustain  the  pressure  ghaD, 
and  also  an  additional  pressure  due  to  its  change  of 
position ;  but  this  latter  pressure  is  infinitely  small. 
Hence  we  may  regard  the  normal  pressure  of  the 
fluid  upon  this  elementary  plane  as  the  same,  what- 


HYDROSTATICS.  197 

ever  position  it  may  assume  about  any  one  of  its 
points  considered  as  fixed,  and  as  equal  to  the  pro- 
duct ghaD ;  that  is,  to  the  weight  of  a  vertical  column 
of  the  fluid  having  the  element  in  its  horizontal  position 
as  a  base,  and  its  distance  from  the  free  surface  of  the 
fluid  as  its  altitude. 

To  apply  this  principle  to  the  case  of  a  surface  of 
finite  extent  pressed  by  the  weight  of  a  fluid,  con- 
ceive the  surface  to  be  divided  into  an  infinite  num- 
ber of  infinitely  small  parts ;  and  let  the  areas  of 
these  parts,  supposed  plane,  be  denoted  by  a,  a  ,  a", 
etc.,  and  their  distances  below  the  surface  of  the  fluid 
by  h,  h\  K\  etc. :  then  the  normal  pressure  upon  these 
surfaces  will,  as  has  just  been  shown,  be  expressed  by 
the  products  ghaD,  gh'a'D,  gh"a"D,  etc. ;  and  hence, 
denoting  the  sum  of  these  elementary  pressures  or 
the  total  pressure  b}^  P,  we  have 

P  zzz  ghaD  -\-gJi' a' D-{-gh" a" D  -\- etc. 
=   Dglah. 

But  denoting  the  sum  of  the  elements  a,  a,  d\  etc., 
that  is,  the  area  of  the  surface  pressed  by  A,  and  the 
distance  of  its  centre  of  gravity  below  the  surface  of 
the  fluid  by  h^  we  have  (  Statics,  Art.  45 ), 

Ah,  =:  ah  -\-  a'h'  -\-  a'h'  -\-  etc.   =  ^ah  : 

consequently, 

P  =  gh,AD. 

Whence  it  appears  that  the  pressure  of  a  fluid  on  any 
submerged  surface  is  equal  to  the  weight  of  a  column  of 
the  fluid  lohose  base  is  equal  to  the  area  of  the  surface, 
and  whose  height  is  equal  to  the  depth  of  the  centre  of 


198  MECHANICS. 

gravity/  of  the  surface  below  the  free  surface  of  the  fluid. 
The  submerged  surface  may  obviously  belong  either 
to  the  surface  of  the  vessel  containing  the  fluid,  or 
to  the  surface  of  an  immersed  body. 

The  student  will  observe,  that  by  "  the  pressure 
of  a  fluid"  on  a  surface,  is  meant  the  sum  of  all  the 
elementary  pressures.  When  the  surface  is  plane, 
this  sum  is  equal  to  the  resultant  of  the  elementary 
pressures. 

When  in  the  case  just  considered,  the  vessel  con- 
tains several  superimposed  strata  of  fluids  of  differ- 
ent densities,  the  pressure  exerted  by  each  fluid  upon 
the  surface,  either  directly  or  by  means  of  the  other 
fluids  below  it,  must  be  determined  separately  :  the 
sum  of  these  j)artial  pressures  will  be  the  total 
pressure  required. 

11.  Centre  of  pressure. 
The  point  of  application  of  the  resultant  of  all  the 
elementary  pressures  exerted  by  a  fluid  upon  an 
immersed  surface  is  called  the  centre  of  pressure.  When 
the  surface  is  plane,  the  pressures  being  all  parallel 
to  each  other,  their  resultant  will  be  equal  ta  their 
sum.  If  the  surface  is  also  horizontal,  the  pressures 
being  then  equal  to  each  other  as  well  as  parallel, 
the  centre  of  pressure  wdll  coincide  with  the  centre 
of  gravity  of  the  plane ;  but  when  the  plane  is  in- 
clined, the  centre  of  pressure  will,  since  the  pressure 
increases  with  the  depth,  lie  below  the  centre  of 
gravity.  We  shall  consider  only  the  case  of  plane 
surfaces. 


I 


HYDROSTATICS.  199 

Let  the  plane  AYEF  [Fig.  7]  represent  the  free 
surface  of  the  fluid;  A'CD,  the  plane  surface  whose 
centre  of  pressure  is  sought ;  and  AY,  the  line  in 
which  the  former  is  intersected  by  the  latter  pro- 
duced. Let  AX,  drawn  in  the  plane  A'CD  perpen- 
dicular to  AY,  be  taken  as  the  axis  of  x,  and  AY  as 
the  axis  of  y.  Employing  the  same  notation  as  in 
Art.  10,  the  pressure  upon  any  element  of  the  sur- 
face, that  at  0.'  for  example,  will  be  expressed  by 
the  product  ghaD ;  and  denoting  the  co-ordinates  of 
O'  hj  X  and  y,  the  moments  of  this  element,  referred 
to  the  axes  AY  and  AX,  will  be  expressed  by  the 
products  ghaDx,  ghaDy.  Also  the  total  pressure 
upon  the  plane,  or  the  resultant,  will  be  expressed 
[Art.  10]  by  ghAD.  Whence  denoting  the  co-ordi- 
nates of  the  point  of  application  (not  represented 
in  the  figure)  of  the  resultant,  that  is,  of  the  centre 
of  pressure,  by  :r^^,  y^^,  we  shall  have  (Statics,  Art.  43), 

IghaDx  '^hax 

^"  =  S^  =  0--. H 

But  denoting  the  angle  XAF,  which  the  plane  of 
A'CD  makes  with  the  surface  of  the  fluid,  by  t,  and 
the  abscissa  BO  of  the  centre  of  gravity  0  of  A'CD, 
by  x^,  we  have 

li   ^L  X  sin  t    and  h,  =:  x,  sin  t ; 

and  by  substitution  in  (a)  and  (b),  we  get 

{d) 


— 

Irtz-  sin  t 



2ax^' 

XjA  sin  t 
laxy  s\nt 

x^A  ' 

'I.axy 

x,A  sin  t 

^/•^ 

200  MECHANICS. 

If  the  point  A  for  which  AAj  =  h,  be  taken  as 
the  origin  of  co-ordinates,  and  A'Y'  parallel  to  AY 
as  the  axis  of  ordinates,  then  BO'  will  be  equal  to 
{h-\-x),  and  we  shall  have 

h   z=   (  6  4"  ^  )  sill  ^     and  h,  =   (  6  -f-  x^ )  sin  t  ; 

and  hence 


z=z 

la(h  +  xjx  sin  t 

= 

lafbx  +  x') 

// 

(b  +  Xi)Asmt 

bA-^XjA    ' 

0/ 

= 

la(b  +  x)y  sin  t 
(h  +  Xj)A  sin  t 

= 

^a(by  +  xy) 
bA  +  XjA    ■ 

ie) 


if) 


Since  the  co-ordinates  x^.,  y^^,  are  independent  of 
the  angle  t,  the  centre  of  pressure  must  be  the  same 
for  all  positions  which  the  given  plane  may  be  made 
to  assume  by  revolving  it  about  AY  as  an  axis,  as 
long  as  AA'  retains  the  same  value. 

If,  in  equations  (e)  and  (/),  Ave  make  b  infinite,  we 
get 

^n  =  -^    and  y„  —  -J-, 

in  which  the  values  of  x  ^  and  y^^  are  the  expressions 
of  the  co-ordinates  of  the  centre  of  gravity  of  the 
surface.  Whence  it  appears  that  as  the  plane  is  de- 
pressed in  the  fluid,  its  centre  of  pressure  constantly 
approaches  its  centre  of  gravity. 

The  quantity  h  may  also  be  rendered  infinite  by 
causing  the  plane  to  revolve  about  the  axis  A'Y',  till 
it  becomes  parallel  to  the  surface  of  the  fluid;  in 
which  position  we  already  know  that  the  two  centres 
coincide. 


HYDROSTATICS.  201 

12.  Applications  of  the  above  formulcn. 

Let  the  plane  surface  whose  centre  of  pressure  is 
Bought,  be  a  rectangle  ABCD  [Fig.  8],  which  has  one 
of  its  sides  AB  in  the  surface  of  the  fluid.  Let  the 
origin  of  co-ordinates  be  taken  at  A"  the  middle  of 
AB  :  we  have,  equations  (c)  and  {d), 

-     —  and  y,,  =  -^. 


"  ~    Ax, 

Since  the  axis  A 'X  bisects  the  rectangle,  the 
centre  of  pressure  will  lie  in  that  axis,  and  we  shall 
have  y^^  —  0.  To  determine  x^^,  we  remark  that  the 
numerator  ^ax^  is  the  moment  of  inertia  of  the 
rectangle  with  respect  to  the  axis  AY.  To  determine 
this,  we  have  {Dynamics,  Art.  45,  equation  (^)), 

in  which  a  denotes  the  distance  between  the  two 
axes  with  respect  to  which  the  moments  of  inertia 
are  taken. 

Employing  the  same  notation  as  in  Dynamics, 
Art.  46,  2\  we  have 

2  rj^m  z=  M  — ,   and  a  =  -^^  =  -^  : 


hence  we  get 

=  m'i+m';  =  3iI 

and  hence 

■Lax'  =  M%. 

O 

We  also  have 

Az,^m\: 

m\ 

consequently 

rr        —             ^     —     97 

26 


202  MECHANICS. 

If  the  upper  side  AB  of  the  rectangle  is  beneath 
the  surface  of  the  fluid,  and  parallel  to  it  as  in 
figure  9,  then  denoting  A"H  by  c,  we  have 

and  proceeding  as  above,  we  get 

We  also  have 

consequently  (equation  (c) ), 

_  6c'^  +  6cl  +  2/2 
^"  ~        6c +  31 

where  d  =z  A'^-\-AD  =  c-\-l. 

13.  Let  the  surface  whose  centre  of  pressure  is 
sought,  be  a  triangle  :  and  first  let  one  of  the  vertices 
of  the  triangle  be  in  the  free  surface  of  the  fluid; 
and  let  the  opposite  side  BC  [Fig.  10],  which  we  as- 
sume as  its  base,  be  parallel  to  the  surface.  If  we 
denote  the  altitude  of  the  triangle  by  h,  its  moment 
of  inertia,  referred  to  the  axis  drawn  in  the  plane  of 
the  triangle  through  its  centre  of  gravity  and  parallel 

to  its  base  BC,  will  be  ^Tg."^  Hence  in  equation  (c) 

*To  find  the  moment  of  inertia  of  a  triangle  ABC  [Fig.  13]  with 
respect  to  an  axis  DE,  drawn  in  the  plane  of  the  triangle,  through  one 
of  its  vertices  A  parallel  to  the  opposite  side  BC,  let  the  height  of  the 
triangle  be  denoted  by  h  and  its  base  by  b,  and  let  h  be  divided  into 


HYDROSTATICS.  203 

2  0x2 


X,,    z=z 


"  ~    Axj 


we  have  for  the  case  in  question, 

and  Ax,  =  p.M; 

n  equal  and  indefinitely  small  parts.  Through  each  point  of  division 
let  lines  be  drawn  parallel  to  BC,  dividing  the  triangle  into  trapezoids. 

The  common  height  -  of  the.se  trapezoids  being  indefinitely  small,  they 

may  be  regarded  as  rectangles ;  their  bases  being 

b   2b    36  nb 

-,  — ,  — , —  or  6, 

n     n      n  n 

and  their  areas 

6^1    2bh    Sbh  bh 

n-      n^       n-  n 

But  the  distances  of  the  rectangles  from  the  axis  DE  are 

h    2h    8h  nh 

-,  — ,  — , —  or  /I  : 

n     n     n  n 

hence  the  moment  of  inertia  of  the  triangle  is 
bh?   ,   2%h^     ,   3'^6/i3     , 

^'(1  +  2^+3^+ n\ 

But  if  in  the  identical  equation 

[x  —  iy  —  x*  — 4x^  +  6x^  — 4x+l, 

we  substitute  successively  for  x  each  of  the  natural  numbers   i,  2,  3, 
n,  and  proceed  as  in  Dynamics,  Art.  46,  note,  we  shall  get 

s'"  =  r  +  2'  +  3' »=  =  ^  +  Y+T'- 

whence  the  moment  of  inertia  sought  becomes 
bh^ 


bh?  /n*     iVL-L.  !!l\ 
^VT""     2     '   T/' 


and  this,  when  n  is  infinite,  becomes 

-r-  or  31 
4 

since  the  mass  31  is  proportional  to  the  area. 


bh^        ^.h' 


204  MECHANICS. 

h?     ■     i¥ 
11  18  "^    9 

and  hence  ^n  =  — j^ — 

=  Ih. 
But  the  centre  of  pressure  must  lie  in  th  tine  A'D, 
drawn  from  the  vertex  to  the  middle  of  the  base. 
Hence  if  we  lay  off  from  A  on  the  axis  AX  a  dis- 
tance equal  to  ^h,  and  through  the  extremity  of  this 
co-ordinate  draw  a  line  parallel  to  AY,  the  point  in 
which  it  intersects  A'D  will  be  the  point  sought. 
This  point,  it  is  evident,  will  be  at  a  distance  from 
A'  equal  to  |A'D. 

When  one  of  the  sides  of  the  triangle  lies  in  the 
free  surface  of  the  fluid,  as  in  figure  11,  we  have 

18    ""^ 


Hence  in  this  case  the  centre  of  pressure  is  the 
middle  point  of  the  line  drawn  from  A'  to  D,  the 
middle  of  BC. 

When  the  triangle  is  beneath  the  surface  of  the 
fluid,  with  one  of  its  sides  BC  horizontal  as  in  figure 

To  find  the  moment  of  inertia  of  the  triangle  referred  to  the  axis 
drawn  through  the  centre  of  gravity  parallel  to  DE,  we  have  {Dynamics, 
Art.  45), 

I.r'm  —  lr;m  +  a''3I, 

or                                    2r/m  z=:  2r'-m  — 0^31: 
whence  Ir^m  =  in| i^)' ^^ 


HYDROSTATICS.  205 

12,  then  denoting  by  c  the  distance  of  BC  from  the 
axis  AY,  we  have 

or,  denoting  the  distance  of  the  vertex  A'  from  the 
axis  AY  by  d,  we  have 


i/M+i/, 


"-  M[c  +  l{d-c)) 

~~         2(d  +  2c)      ' 

from  which  the  point  required  can  be  immediately 
found. 

For  the  determination  of  the  centres  of  pressure 
of  surfaces  in  general,  recourse  must  be  had  to  the 
Integral  Calculus. 

14.  Of  the  resultant  of  the  pressures  exerted  by  a  fluid 
upon  the  surface  of  a  body  either  wholly  or  partially 
immersed. 

In  the  preceding  articles,  we  have  determined  the 
sum  of  the  elementary  pressures  exerted  by  a  fluid 
upon  the  surface  of  the  vessel  containing  it,  and 
also  upon  any  surface  immersed  in  the  fluid ;  and 
remarking  that  when  the  surface  is  plane,  the  re- 
sultant of  the  elementary  pressures  is  equal  to  their 
sum,  we  have  investigated  for  this  case  the  general 
expressions  of  the  co-ordinates  of  the  point  of  appli- 
cation of  the  resultant,  or  the  centre  of  pressure  of 
the  plane.  To  determine  the  resultant  of  the  ele- 


206  MECHANICS. 

mentary  pressures  in  any  case  whatever,  as,  for 
example,  the  resultant  of  the  pressures  upon  any 
portion  of  the  surface,  supposed  to  be  curved,  of  an 
immersed  solid,  requires  the  aid  of  the  Integral 
Calculus.  We  shall  consider  only  the  case  in  which 
it  is  desired  to  find  the  resultant  of  the  pressures  exerted 
upon  the  entire  surface  of  an  immersed  body. 

Let  AX,  AY,  AZ  [Fig.  14]  be  the  axes  of  a  system 
of  rectangular  co-ordinates,  the  plane  XAY  coincid- 
ing with  the  free  surface  of  the  liquid.  Let  OQM  be 
the  body  in  question,  and  m  any  point  of  its  surface ; 
and  let  the  co-ordinates  of  m  be  denoted  by  x,  ?/, 
and  z.  Let  an  infinitely  small  portion  of  the  surface 
at  the  point  m  be  denoted  by  a,  and  the  pressure 
upon  the  unit  of  surface  at  the  depth  z  hy  p  (where 
p  =  gDz)  :  then  the  pressure  sustained  by  the  ele- 
ment a  will  be  expressed  by  pa,  its  direction  mn 
being  normal   to  the  surface   at  m.    Denoting   by 

a,  /3,  y  the  angles  which  mn  makes  with  the  axes  of 
X,  y  and  z  respectively,  we  have  for  the  components 
of  the  pressure  parallel  to  these  axes  {Statics,  Art.  17), 

pa  cos  a,    pa  COS  (3,     pa  cosy. 
Now  let  the  element  a  be  projected  on  the  planes 
YAZ,  XAZ,  XAY;  and  denote  the  projections  by 

b,  b'y  b"  respectively.  The  angle  which  the  plane  of 
the  element  makes  with  the  plane  of  YAZ  is  equal 
to  the  angle  made  by  the  lines  mn,  mm^  (mm^  being 
parallel  to  the  axis  AX),  normals  to  these  planes 
respectively ;  and  the  same  being  true  of  a  referred 
to  the  other  co-ordinate  planes,  we  have 


HYDROSTATICS.  207 

b  -==:  a  cos  a,     b'  :=.  a  cos  /3,     b"  =:  a  cos  y  ; 
or,  multiplying  by  p, 

jjb  =  pa  COS  a,    pb'  =z  pa  cos  (3,    pb"  z=  pa  cosy: 

whence  it  appears  that  the  components  of  the  pres- 
sure pa,  parallel  to  the  axes  of  x,  y,  and  z,  may  be 
expressed  by  the  products  ph,  pb',  pb". 

But  the  perpendiculars  by  which  the  contour  of 
the  element  a  is  projected  upon  the  plane  YAZ  will 
determine  upon  the  surface  of  the  body,  at  some 
point  m',  another  element  a,  which  will  have  the 
same  projection  b  upon  that  plane  as  the  element  a ; 
and  since  these  elements  a  and  a'  are  both  at  the 
same  distance  from  the  surface  of  the  fluid,  the 
pressure  upon  a  will  be  expressed  by  pa,  and  its 
component  parallel  to  the  axis  of  x  by  pb.  The  two 
components  of  pa  and  pa',  parallel  to  the  axis  of  x, 
being  thus  equal  to  each  other,  and  also  acting  in  the 
same  line  in  opposite  senses,  will  destroy  each  other. 

The  perpendiculars  by  which  the  contour  of  the 
element  a  is  projected  upon  the  plane  XAZ  will  in 
like  manner  determine  on  the  surface  of  the  body, 
at  some  point  m",  a  third  element  a",  on  which  the 
normal  pressure  is  pa";  and  the  component  of  this 
pressure  parallel  to  the  axis  of  y,  viz.  pb',  will,  as  in 
the  preceding  case,  destroy  the  second  horizontal 
component  pb'  of  the  pressure  pa.  In  this  manner 
will  the  horizontal  components  of  the  pressures 
exerted  by  the  fluid  upon  all  the  elements  of  the 
body,  destroy  each  other. 


208  MECHANICS. 

15.  It  remains  to  find  the  resultant  of  the  vertical 
components.  The  perpendiculars  by  which  the  con- 
tour of  a  is  projected  on  the  plane  XAY  determine 
on  the  surface  of  the  body  at  m"  another  element 
a" ,  which  will  have  the  same  horizontal  projection 
h"  as  a.  These  perpendiculars  will  form  the  convex 
surface  of  a  vertical  cylinder,  terminated  at  the 
surface  of  the  body  by  the  elements  a  and  a"\  and 
having  for  its  horizontal  section  the  projection  h" 
common  to  these  two  elements.  The  vertical  height 
mm'"  of  this  cylinder  is  the  distance  between  these 
two  elements;  and  denoting  this  by  /,  its  volume 
will  be  expressed  by  h"l.  The  normal  pressures  upon 
the  bases  a  and  a"  will  be  pa  and  pa"  {p  being 
the  unit  of  pressure  at  the  depth  of  m") ;  and  the 
vertical  components  of  these  pressures  will  be  ph" , 
ph'\  the  body  being  supposed  to  be  completely  im- 
mersed. The  resultant  of  these  components  wall  be 
ph"  —  ph"  or  {p  — p')b\  a  being  supposed  the  element 
most  distant  from  the  plane  XY.  But  ^  =  gDz ; 
and  denoting  by  z'  the  vertical  co-ordinate  of  m'", 
p    —  gDz'  :  wdience 

P—P'  =  gl>\-  —  ^')  =  gDl, 
and  {p  —  p')  b"  =  gb"  Dl, 

that  is,  the  elementary  cylinder  is  pressed  vertically 
upwards  by  a  force  equal  to  the  w^eight  of  a  mass  of 
the  fluid  w^iose  volume  is  the  same  as  that  of  the 
cylinder. 

The  same  being  true  of  all  the  vertical  elementary 
cylinders  composing  the  body,  the  whole  body  will  be 


HYDROSTATICS.  209 

urged  vertically  upioards  by  a  force  equal  to  the  weight 
of  a  mass  of  the  fluid  whose  volume  is  the  same  as  that  of 
the  body.  Denoting  the  volume  of  the  body  by  V,  this 
force,  the  resultant  of  the  pressures  exerted  by  the 
fluid  upon  its  entire  surface,  will  be  expressed  by 
the  product  gDV.  Its  direction  will  pass  through  the 
centre  of  gravity  of  the  mass  of  fluid  displaced  by  the  body. 
For  denoting  by  x^,  y,  the  distances  of  its  point  of 
application  from  the  planes  YAZ,  XAZ,  we  have 

_     ^gDb"lx  _     I,gDb"ly 

""'  ~   gDV    ^"""^  y'  -   gDv'  '■ 

in  which  equations  the  second  members  are  identical 
Avith  the  expressions  of  the  distances  of  the  centre 
of  gravity  of  the  displaced  fluid  from  the  co-ordinate 
planes  respectively. 

In  the  case  of  a  body  partially  immersed,  the  re- 
sultant of  the  vertical  components  is  equal  to  the 
weight  of  the  mass  of  fluid  displaced  by  the  immersed 
part  of  the  body,  and  its  direction  passes  through  the 
centre  of  gravity  of  the  mass  thus  displaced. 

In  the  case  in  which  the  fluid  is  composed  of  hori- 
zontal and  homogeneous  strata  of  different  densities, 
we  conceive  the  several  strata  to  be  produced  within 
the  body,  forming  a  mass  of  fluid  similar  to  that  sur- 
rounding the  body,  and  of  the  same  volume  as  the 
body  or  its  immersed  part.  The  resultant  of  the 
pressures  will  evidently  be  equal  to  the  weight  of 
this  mass,  and  will  pass  through  its  centre  of  gravity. 


27 


210  MECHANICS. 

16.   Conditions  of  equilibrium. 

The  conditions  of  equilibrium  of  a  solid  body 
immersed  in  a  fluid  obviously  are,  1st,  that  the  centre 
of  gravity  of  the  body,  and  that  of  the  fluid  displaced 
by  the  body,  shall  be  in  the  same  vertical  line ;  and  2d, 
that  the  iveight  of  the  displaced  fluid,  that  is,  the  resultant 
of  the  pressures  exerted  by  the  fluid  upon  the  body,  shall 
be  equal  to  the  weight  of  the  body. 

When  the  body  is  homogeneous,  and  entirely 
immersed,  the  two  centres  of  gravity  necessarily 
coincide,  and  the  first  condition  is  always  fulfilled ; 
and  in  order  that  the  second  condition  may  be 
satisfied,  it  is  only  necessary  that  the  densities  of 
the  fluid  and  of  the  immersed  body  should  be  equal. 
If  the  densities  are  not  equal,  the  body  will  rise  or 
sink,  according  as  its  density  is  less  or  greater  than 
that  of  the  fluid.  The  body  may  be  kept  at  rest  in 
the  first  case  by  means  of  an  inextensible  thread,  the 
upper  extremity  of  which  is  attached  to  the  body, 
and  the  lower  to  the  bottom  of  the  vessel ;  in  the 
second  case,  by  a  similar  thread,  the  lower  extremity 
of  which  is  attached  to  the  body,  and  the  ujDjier  to 
a  fixed  point.  In  either  case,  the  tension  of  the 
string  will  be  equal  to  the  difierence  between  the 
weights  of  the  body  and  the  fluid  displaced.  If,  in 
the  latter  case,  the  string  were  attached  to  one  of 
the  scales  of  a  balance,  and  the  body  were  thus 
weighed  while  immersed  in  the  liquid,  it  is  evident 
that  it  would  weigh  less  than  in  air,  the  loss  of  weight 
being  equal  to  the  weight  of  the  fluid  displaced. 


HYDROSTATICS.  211 

From  this  it  appears  that  two  bodies  which  are  in 
equilibrium  with  each  other  when  weighed  by  an 
exact  balance  in  air,  or  any  other  fluid,  have  really 
different  weights,  unless  their  volumes  are  equiva- 
lent ;  the  body  whose  volume  is  the  greater  obviously 
having  the  greater  weight.  To  obtain  the  true  weight 
of  a  body,  it  should  be  weighed  in  a  vacuum. 

17.  The  resultant  of  the  pressures  exerted  hy  a  fluid 
upon  the  whole  interior  surface  of  the  vessel  containing  it. 

The  normal  elementary  j)i'essures  being  each  re- 
solved into  three  components  at  right  angles  to  each 
other,  two  horizontal  and  the  third  vertical  as  in 
article  14,  it  can  be  shown,  as  in  that  article,  that 
the  resultant  of  the  horizontal  pressures  is  zero.  The 
horizontal  components  being  in  equilibrium  among 
themselves,  the  resultant  sought  must  be  the  result- 
ant of  the  vertical  components ;  and  this,  it  can  be 
shown,  as  in  article  15,  is  equal  to  the  weight  of  the 
mass  of  fluid,  and  passes  through  its  centre  of  gravity. 

This  resultant  of  the  vertical  pressures  must  be 
carefully  distinguished  from  the  pressure  sustained 
by  the  horizontal  bottom  of  the  vessel,  from  which 
it  differs,  except  in  the  single  case  in  which  the 
vessel  is  a  right  prism  or  cylinder.     (See  Art.  7.) 

From  the  above,  it  appears  that  a  vessel  contain- 
ing a  fluid,  placed  upon  a  horizontal  plane,  exerts 
upon  the  plane  a  pressure  equal  to  the  sum  of  the 
weights  of  the  vessel  and  fluid. 

Since,  in  the  case  of  a  vessel  containing  a  fluid, 
the  horizontal  pressures  destroy  each  other,  they  can 


212  MECHANICS. 

have  no  tendency  to  communicate  motion  to  the 
vessel.  If,  however,  an  aperture  be  made  in  the  side 
of  the  vessel  at  any  point,  and  the  fluid  be  permitted 
to  escape,  the  uncompensated  pressure  at  the  point 
directly  opposite  will  tend  to  impart  motion  to  the 
vessel,  in  the  direction  opposite  that  in  which  the 
fluid  escapes. 

18.   Of  the  equilibrium  of  floating  bodies. 

In  order  that  a  floating  body  may  be  in  equili- 
brium, it  is  necessary  [Art.  16],  1st,  that  the  centre 
of  gravity  of  the  body,  and  that  of  the  fluid  dis- 
2)laced,  should  be  in  the  same  vertical  line  ;  and,  2d, 
that  the  weight  of  the  body,  and  that  of  the  displaced 
fluid,  should  be  equal  to  each  other. 

Denoting  the  volume  of  the  body  supposed  to  be 
homogeneous,  by  V ;  the  volume  of  its  immersed 
part,  that  is,  the  volume  of  the  fluid  displaced,  by 
V,  and  the  densities  of  the  body  and  fluid  by  D 
and  D',  we  have,  by  the  second  condition, 

VDg  =   VD'g, 
or  V   :    V    ::    D'    :    D; {g) 

that  is,  the  volume  of  the  entire  body,  to  the  volume  of  the 
immersed  part,  as  the  density  of  the  fluid,  to  the  density 
of  the  body.  Hence  the  determination  of  the  j^ositions 
of  equilibrium  of  a  floating  body  is  reduced  to  the 
following  problem  of  pure  geometry,  viz  :  To  divide 
the  body  by  a  plane,  so  that  its  entire  volume  shall  be  to 
the  volume  of  one  of  its  segments  in  a  given  ratio,  and  the 
dividing  plane  itself  be  perpendicular  to  the  line  joining 
the  centre  of  gravity  of  the  body  and  that  of  the  segment. 


HYDROSTATICS.  213 

111  any  given  case,  these  conditions  must  be  ex- 
pressed by  equations,  the  solutions  of  which  deter- 
mine the  positions  of  the  cutting  plane,  and  hence 
the  positions  of  equilibrium.  We  will  take  as  an  ex- 
ample the  case  of  a  right  triangular  prism,  in  the 
position  in  which  the  axis  is  horizontal,  and  conse- 
quently parallel  to  the  cutting  plane.  In  this  case, 
it  is  evident  that  the  centre  of  gravity  of  the  entire 
prism  coincides  with  the  centre  of  gravity  of  that 
section  of  the  prism  which  bisects  its  axis  at  right 
angles,  and  also  that  the  centre  of  gravity  of  the 
immersed  part  of  the  prism  coincides  with  that  of 
the  immersed  part  of  the  section.  The  problem  is 
thus  reduced  to  finding  the  position  of  equilibrium 
of  this  section.  Let  ABC  [Fig.  14']  be  the  section,  and 
MN  the  straight  line  in  which  it  is  intersected  by  the 
cutting  plane  or  level  of  the  fluid,  only  one  vertex 
C  being  immersed.  Let  the  sides  of  the  triangle  ABC, 
which  are  opposite  the  angles  A,  B,  and  C  respec- 
tively, be  denoted  by  a,  h  and  c ;  and  let  the  unknown 
sides  CN  and  CM  of  the  triangle  CMN  be  denoted 
by  X  and  y  :  we  have,  by  trigonometry,  the  area  of 

the  triangle 

ABC  =  Jrt6smC, 

and  that  of  the  triangle 

MNC  =  JarysinC. 

But  the  entire  prism  and  its  immersed  part  ari  t;j 
each  other  as  the  triangles  ABC,  MNC  respectively  : 
hence,  denoting  the  densities  of  the  fluid  and  body 
by  1  and  D  respectively,  we  have  (Proportion  (g)  ) 


214  MECHANICS. 

ABC    :    MXC    ::    1    :    D; 
and  substituting  for  ABC,  MNC  the  values  just  ob- 
tained, and  suppressing  the  factor  sin  C,  we  have 

xy  =z  ahD (A) 

Again,  let  the  points  G  and  G',  the  centres  of  gravity 
of  the  triangles  ABC,  MNC,  be  determined  {Statics, 
Art.  46,  5")  :  then,  in  order  to  an  equilibrium,  it  is 
necessary  that  the  straight  line  GG  should  be  per- 
pendicular to  the  line  MN.  But  since  GD  =  -|CD 
and  G'K  =  JCK,  the  straight  line  DK  is  parallel  to 
GG';  and  hence  DK  also  must  be  perpendicular  to 
MN,  and  consequently  the  straight  lines  DM  and  DN 
must  be  equal  to  each  other.  Reciprocally  if  DM  = 
DN,  the  straight  line  DK,  and  hence  its  parallel  GG  , 
will  be  perpendicular  to  MN.  Therefore  in  order 
that  the  straight  line  which  joins  the  two  centres  of 
gravity  G  and  G'  may  be  perpendicular  to  the  inter- 
section MN  of  the  cutting  plane,  it  is  necessary  and 
sufficient  that  the  two  lines  DM  and  DN  should  be 
equal  to  each  other.  To  find  the  values  of  these 
lines,  let  the  straight  line  CD  be  denoted  by  A,  and 
the  angles  DCM,  DCN  by  p  and  q  :  then  from  the 
triangle  DCM,  we  get 

DM^  =  CD^  +  Or  — 2CD.CMcosDCM 
=z  Jr  -\-  1/-  —  2  hi/  Gosp  ; 
and  from  the  triangle  CDN, 

DN'  =  /r  +  z"  — 2/ixcos  j; 
and  hence  we  have 

3/^  —  2  hy  cos  p  =  x"  —  2  Ax  cos  q .  (/) 


HYDROSTATICS.  215 

This  equation,  and  equation  (A),  express  the  two 
conditions  of  the  problem,  and  serve  to  determine 
the  unknown  quantities  x  and  y.  Eliminating  y  from 
the  first  equation,  and  reducing,  we  get 

x'—2h  cos  qx'  +  2  abDh  cospx  —  a'bW  =  0 {j) 

The  value  of  x  being  deduced  from  this  equation, 
the  corresponding  value  of  y  becomes  known  from 
equation  (A). 

But  it  is  shown  in  the  elements  of  algebra,  that 
every  equation  of  an  even  degree,  whose  last  term 
is  negative,  has  at  least  two  real  roots,  the  one 
positive,  the  other  negative.  Equation  (j)  will  then 
always  have  two  such  roots,  the  two  remaining  roots 
being  either  real  or  imaginary.  If  the  four  roots  of 
equation  (j)  are  real,  then  according  to  the  rule  of 
Descartes,  three  of  them  must  be  positive  and  one 
negative.  For  inserting  in  equation  (j)  the  term 
whose  co-efficient  is  zero,  by  placing  after  the  term 
in  x^  the  term  +  Ox',  we  perceive  that  there  are 
three  variations  and  one  permanence.  But  since  the 
values  of  :r  and  ?/are  essentially  positive,  the  negative 
values  of  equation  (j)  must  be  rejected.  Nor  can  any 
value  of  X  be  admitted  which  is  greater  than  a ;  or 
which,  though  smaller  than  a,  will  give  for  y  a  value 
greater  than  b.  It  thus  appears  that  in  the  case  under 
consideration,  in  which  only  one  vertex  is  immersed, 
there  are  at  the  most  only  three  positions  of  equili- 
brium. 

The  case  in  which  the  vertex  C  is  without  the 
fluid,  and  the  vertices  A  and  B  are  immersed,  is 


216  MECHANICS. 

easily  reducible  to  tlie  one  just  considered.  For  the 
centres  of  gravity  of  the  triangle  ABC,  and  its  two 
parts  CMN,  MNBA,  lie  in  the  same  straight  line  GG' 
produced ;  and  hence  GG'  produced  is  still  the  line 
which  joins  the  centre  of  gravity  of  the  body  and 
that  of  the  immersed  segment.  Expressing  analyti- 
cally the  condition  that  this  line  is  perpendicular  to 
MN,  as  we  have  done  above,  we  get  an  equation 
identical  with  equation  (i).  Moreover  the  proportion 

ABC    :    MNBA    ::    1    :    D, 
gives  CMN    :    ABC    ::    l—B    :    1; 

whence  substituting  for  CMN  and  ABC  the  values 
found  above,  we  get 

3-3/  =  (1  -  D)ab, 

the  equation  which  takes  the  place  of  the  equation 
xy  =  ahD  of  the  preceding  case.  If  we  employ  this 
to  eliminate  y  from  equation  (2),  we  get 

x'—11i(:Q^qx'-[-1ahh{\—D)Qo>px  —  arK-{l  —  Df  =  0;  [Ic) 

which  differs  from  equation  (j)  only  in  having  the 
factor  (1  —  B)  in  the  j^lace  of  D. 

Reasoning  as  above,  then,  it  can  be  shown  that 
also  in  the  case  in  which  the  two  vertices  A  and  B 
are  immersed  and  C  without  the  fluid,  the  positions 
of  equilibrium  cannot  exceed  three.  By  considering 
thus  for  each  vertex  the  case  in  which  it  alone  is 
immersed,  and  that  in  which  it  alone  is  without  the 
fluid,  we  can  determine  all  the  possible  horizontal 
positions  of  equilibrium  of  the  given  prism.  From 
the  preceding  discussion,  it  appears  that  the  number 
of  these  positions  can  never  exceed  eighteen. 


HYDROSTATICS.  217 

19.  Let  us  now  suppose  the  triangle  ABC  [Fig.  15] 
to  be  isosceles,  AC  and  BC  being  the  equal  sides  and 
C  the  immersed  vertex.  In  this  case,  the  line  CD 
bisects  the  side  AB  at  right  angles ;  and  from  the 
triangle  ADC,  we  get 

h"  zzz  d"  —  |c^     and  a  cos  q  =:  h  ; 
and  hence  2  h  cos  q  =  —  =  — ^ . 

To  adapt  equations  (A)  and  (?)  to  this  case,  in  equa- 
tion (h)  we  make  b  —  a,  and  thus  get 

xi/  —   aW  ; (/) 

and  in  equation  (z)  we  put  p  =  q,  and,  substituting 
for  2  h  cos  q  the  value  just  found,  get 

These  equations  are  satisfied  by  taking  x  =  y  = 
aVD.  Moreover  since  i)<l,  and  hence  as/B<^a, 
these  equal  values  are  admissible  :  consequently  one 
position  of  equilibrium  is  that  in  which  the  two 
sides  CM  and  CN  are  equal  to  each  other,  the  side 
AB  being  horizontal. 

Su|)j)ressing  the  factor  y  —  x  in  equation  (m),  we 
have,  to  determine  the  remaining  values  of  x  and  ?/, 
the  equations 

%y  ^=  Dd^     and  y  -\-x  z=  — . 

From  these  we  find 


^  =  Va (") 

and  ^^4a^-c-^+V(4«^-e^J--16Pa^ ^^^ 

28 


218  MECHANICS. 

When  these  values  of  x  and  y  are  real,  and  each  less 
than  «,  they  determine  two  new  positions  of  equi- 
librium in  which  the  base  AB  is  inclined. 

In  the  case  in  which  the  side  AB  is  immersed 
and  the  vertex  C  without  the  fluid,  we  have  only  to 
substitute  in  the  equations  of  the  preceding  case 
(1  —  D)  in  place  of  D. 

20.  Let  the  triangle  ABC  be  equilateral. 

In  this  case,  a  =  b  =  c;  and,  as  in  the  case  of 
the  isosceles  triangle,  there  is  one  solution  in  which 
the  values  of  x  and  y  are  equal  to  each  other,  each 
being  equal  to  aVD.  To  find  the  unequal  values  of 
X  and  y,  we  have  only  to  make  c  =  a  in  equations 
{?i)  and  (o)  :  we  thus  find 


a(S±-\/  9- 

-16D) 

4 

afSn:  V  9- 

-16D) 

ip) 


and  y  =  — '■ ^ (?) 

In  order  that  these  values  may  be  real,  and  each 
less  than  a,  we  must  have 

16I><9    or  i><T%, 


and  '    V9  —  IQD  <  1     or  Z>  >  /^. 

Without  these  limits,  the  values  of  x  and  y  become 
either  imaginary  or  greater  than  a. 

To  find  the  values  of  x  and  y  in  the  case  in  which 
the  side  AB  is  immersed  and  the  vertex  C  without 
the  fluid,  we  substitute  in  the  equation,  x  —  y  = 
as/B  ',  and  in  equations  (p)  and  (^),  (1  —  JJ)  for  B  : 
we  thus  get 


HYDROSTATICS.  219 


af3±Vl6Z>  — 7j 
and  X  =  -; 


and  Tj  z=      '     ^     ^ . 

Here,  in  order  that  x  and  y  may  be  real  and  less 
than  a,  we  must  have 

16i>  >  7     or  Z>  >  rV, 
and  16Z>  — 7<1     or  D  <:-^\, 

21.  A  right  prism  may  be  in  equilibrium  in  a 
fluid  when  its  axis  is  vertical,  as  well  as  horizontal. 
In  this  case,  its  bases  are  parallel  to  the  free  surface 
of  the  fluid ;  and  since  either  base  may  be  immersed, 
there  are  evidently  two  such  positions  of  equilibrium. 

Since  the  centre  of  gravity  of  the  body,  and  that 
of  the  fluid  displaced,  are  necessarily  in  the  same 
vertical  line,  the  first  condition  of  equilibrium  is 
always  satisfied.  To  determine  the  depth  to  which 
the  body  will  sink,  we  have  only  to  lay  off*  on  one 
of  the  edges  beginning  at  the  immersed  base,  a 
distance  which  shall  be  to  the  entire  length  of  the 
edge,  as  the  density  of  the  body,  to  the  density  of 
the  fluid.  The  volume  of  the  immersed  part  will 
be  to  the  volume  of  the  entire  prism  in  the  same 
ratio,  as  required  by  Art.  18.  The  distance  thus 
laid  off"  will  be  the  depth  sought. 

Solids  of  revolution  have  also  two  positions  of 
equilibrium,  in  which  the  axis  of  form  is  vertical. 
Let  the  solid  be,  for  example,  a  right  cone  :  it  may 
be  in  equilibrium  both  when  the  vertex  is  immersed, 
and  when  it  is  without  the  fluid.  In  the  first  case, 


220  MECHANICS. 

the  part  immersed  is  a  cone  similar  to  the  entire 
cone.  Denoting  the  axis  of  the  entire  cone  by  A,  and 
that  of  the  immersed  cone  by  x,  and  the  densities  of 
the  fluid  and  body,  as  before,  by  1  and  D,  we  have 

and  hence  x  =  A  ^^D. 

When  the  vertex  is  without  the  fluid,  and  the  base 
of  the  cone  immersed,  the  immersed  part  is  a  frus- 
tum of  a  cone ;  and  denoting  its  altitude  by  y,  and 
the  other  magnitudes  as  before,  we  have 

ih-yY    :    h^    ::    1-D    :    l, 

and  y  =  Ti{\—  Vl  —  D). 

In  general,  all  bodies  which  are  symmetrical  with 
respect  to  a  straight  line  or  axis,  have,  like  prisms 
and  solids  of  revolution,  two  positions  of  equilibrium 
in  which  the  axes  are  vertical. 

22.  Stability  of  floating  bodies. 

When  a  floating  body  is  very  slightly  moved  from 
its  position  of  equilibrium,  1st,  it  may  tend  to  return 
to  that  position ;  or,  2d,  it  may  tend  to  depart  still 
farther  from  it ;  or,  3d,  it  may  have  no  tendency  to 
do  either.  In  the  first  case,  the  equilibrium  is  said, 
to  be  stable  ;  in  the  second,  unstable ;  and  in  the  third, 
indifferent.  To  investigate  a  general  rule  for  deter- 
mining whether  the  equilibrium  of  a  floating  body 
is  stable  or  unstable,  would  require  the  use  of  the 
calculus.  We  shall  consider  only  a  particular  case, 
but  one  which  is  of  frequent  occurrence. 


HYDROSTATICS.  221 

This  case  is  that  of  a  body  which  may  be  divided 
by  a  vertical  plane  into  two  parts,  exactly  alike  both 
in  form  and  density.  Let  CED  [Fig.  16]  be  the 
plane,  and  AB  its  intersection  with  the  free  surface 
of  the  fluid  :  also  let  G  be  the  centre  of  gravity  of 
the  body,  and  0  that  of  the  fluid  displaced.  The 
straight  line  GO  will  be  vertical,  and  hence  perpen- 
dicular to  AB.  Let  the  equilibrium  of  the  body  be 
disturbed,  and  in  such  a  manner  that  during  its 
motion  the  section  CED  shall  remain  constantly 
vertical.  Let  the  section,  after  the  disturbance,  be 
represented  in  figure  17,  in  which  the  straight  line 
A'B'  represents  the  free  surface  of  the  fluid,  and  the 
point  0'  the  centre  of  gravity  of  the  fluid  displaced. 
The  forces  which  tend  to  move  the  body  are,  1st, 
its  weight,  which  acts  in  the  direction  GH  vertically 
downwards;  and,  2d,  the  resultant  of  the  vertical 
pressures  of  the  fluid  upon  the  body,  which  is  equal 
to  the  weight  of  the  fluid  displaced,  and  which  acts 
at  the  point  0'  in  the  direction  OH'  vertically 
upwards.  The  point  M,  in  which  GO  and  OH'  inter- 
sect each  other,  is  called  the  metacentre,  and  may  be 
taken  for  the  point  of  application  of  the  upward 
pressure  of  the  fluid.  The  equilibrium  is  obviously 
stable  or  unstable,  according  as  the  straight  line  GE 
tends  to  return  to  the  vertical  position,  or  to  depart 
still  farther  from  it ;  and  an  inspection  of  the  figure 
shows  that, when  the  metacentre  is  above  the  centre  of 
gravity  G,  as  it  is  at  the  point  M,  the  upward  presi^ure 
of  the  fluid  will  tend  to  render  GE  vertical;  and  that 


222  MECHANICS. 

when  it  is  below  G,  as  at  M',  it  will  tend  to  make  GE 
depart  still  farther  from  the  vertical  j^osition.  Thus 
in  tlie  case  in  question,  the  equilibrium  is  stable  or 
unstable,  according  as  the  metacentre  is  above  or  below  the 
centre  of  gravity  of  the  body.  Should  the  metacentre 
coincide  with  the  centre  of  gravity  of  the  body,  there 
would  be  no  tendency  to  motion  either  way,  and  the 
equilibrium  would  be  one  of  indifference. 

23.  Specific  gravity. 

Let  A  and  B  be  two  substances  whose  volumes, 

densities  and  weights  are  denoted  by  V  and  V^,  D 

and  D\  W  and  W\  respectively  :  then  {Statics,  Art. 

41,)  we  have 

W  =   VDg   and   W  =    V'Dg; 

and  hence,  supposing  V  —   V, 

R  —  K 

D'   ~   W 
If,  now,  we  suppose  B  to  be  the  substance  whose 
density  is  assumed  as  the  unit  of  density,  we  have 

w 

D  —  — 

—  m 
This  ratio  ^  of  the  weights  of  equal  volumes  of 

A  and  B,  which  thus  gives  the  density  of  A  in 
terms  of  the  density  of  B,  is  also  called  the  specific 
gravity  of  A,  referred  to  B  as  the  standard  of  specific 
gravity. 

The  specific  gravity  of  a  substance  is  then  found 
by  dividing  the  weight  of  any  volume  of  it,  by  the  weight 
of  an  equal  volume  of  the  substance  whose  specific  gravity 
is  assumed  as  the  unit  of  specific  gravity. 


HYDROSTATICS.  223 

For  solids  and  liquids,  the  standard  substance  is 
pure  water,  usually  taken  at  the  temperature  of  60 ' 
Fahr. ;  for  gases,  pure  and  dry  atmospheric  air  at 
the  temperature  of  32°,  and  under  a  pressure  of  30 
inches  of  mercury. 

24.  Solids. 

To  find  the  specific  gravity  of  a  solid,  let  the  body 
be  weighed  both  in  air  and  in  water  :  then  if  the 
respective  weights  be  denoted  by  w  and  w\  the 
weight  of  a  volume  of  water  equal  to  the  volume 
of  the  solid  will  (Art.  16)  be  expressed  by  w  —  iv  ; 
and  denoting  the  specific  gravity  by  s,  we  shall  have 


w^  ' 


When  the  substance  A  whose  specific  gravity  is 
sought  is  less  dense  than  water,  we  attach  to  it  some 
body  A'  sufficiently  dense  and  heavy  to  sink  it,  and 
weigh  the  two  thus  connected  in  water,  and  also 
weigh  the  body  A  in  air,  and  A'  in  both  air  and 
water.  Let  the  weights  of  A  and  A'  in  air  be  denoted 
by  w  and  w",  and  their  weight  in  water,  when 
connected,  by  iv^^ ;  also  let  the  weight  of  A'  in  water 
be  denoted  by  w'"  :  then  the  loss  of  weight  in  water 
of  the  two,  when  connected,  will  be  exj)ressed  by 
w  -{-  w" —  w''';  but  of  this,  A'  loses  w" —  to'";  and  hence 
A  loses  w  +  w"  —  w'''  —  {w"  —  w'")  z=  w  -\-  w"  —  vT^ 
and  consequently 

w 


224  MECHANICS. 

25.  Liquids. 

To  find  the  specific  gravity  of  a  liquid,  let  some 
suitable  body  be  weighed  in  air  and  in  water,  and 
also  in  the  liquid  whose  specific  gravity  is  sought ; 
and  let  the  respective  weights  be  denoted  by  to,  w 
and  w"  :  then  the  weight  of  a  volume  of  water  equal 
to  the  volume  of  the  body  will  be  expressed  by 
10  —  w\  and  the  w^eight  of  an  equivalent  volume  of 
the  liquid  hy  w — to" ;  and  denoting  as  above  the 
specific  gravity  of  the  liquid  by  5,  we  shall  have 


w  —  w'  ^ 

26.  Hydrometers. 

Let  N  [Fig.  18]  represent  a  sj^mmetrical  hollow 
bulb  of  glass  or  metal,  having  attached  to  it  a  stem 
MR,  whose  lower  extremity  R  is  so  loaded,  that 
when  the  instrument  is  placed  in  a  liquid,  it  will 
float  vertically.  Let  the  upper  portion  of  the  stem 
be  graduated,  commencing  at  the  top  M ;  and  let 
V  denote  the  entire  volume  of  the  instrument,  and 
v'  the  volume,  supposed  constant,  included  between 
any  two  consecutive  divisions  of  the  stem.  Suppose 
the  instrument  to  sink  to  the  point  P  in  water,  and 
to  Q  in  the  liquid  A  whose  specific  gravity  is  sought. 
Let  MP  contain  m,  and  MQ  x  of  the  equal  parts  of  the 
graduated  stem ;  and  let  D  denote  the  density  of  A, 
that  of  water  being  =  1.  Then  the  volumes  of  w^ater, 
and  of  the  liqviid  A  displaced  by  the  instrument, 
wall  be  expressed  by  ^  —  mv  and  v  —  xv  ;  and  the 
weights  of  these  volumes  by  g{v  —  7nv')  and  gB(v  —xv') 


HYDROSTATICS.  225 

respectively ;  and  since  the  weight  of  the  fluid  dis- 
placed is  equal  to  the  weight  of  the  body  immersed, 
and  the  same  body  is  used  for  both  fluids,  we  shall 
have 

gD  {v — XV')  z=  g  {v  —  mv') ', 

V 

and  hence        D  =  '-=^  ^  til  ^  ,, 

V  —  XV'  V 

v' 

the  specific  gravity. 

Here  m  can  be  determined  by  direct  observation ; 
and  then  placing  the  instrument  in  a  liquid  whose 
specific  gravity  s  is  known,  and  observing  the  corre- 
sponding value  of  X  =  x\  we  have  to  determine  ~ 
the  equation 

V 

-j—m 


The  constants  m  and  -^  being  thus  determined,  by 

giving  to  X  the  values  1,  2,  3,  etc.,  the  corresponding 
specific  gravities  can  be  calculated. 

This  instrument  is  called  an  hydrometer.  In  the 
form  just  described,  the  weight  of  the  instrument  is 
constant,  and  the  volume  of  the  part  immersed  vari- 
able. In  another  form,  called  Nicholson's  hydrometer^ 
the  reverse  is  the  case ;  the  volume  immersed  being 
constant,  while  the  mass  may  be  varied  by  the  addi- 
tion of  weights.  In  this  hydrometer,  which  is  repre- 
sented in  figure  19,  a  small  cup  is  attached  to  each 

29 


226  MECHANICS. 

extremity  of  the  stem.  Let  P  be  a  fixed  point  on 
the  stem ;  and  suppose,  that  to  sink  the  instrument 
to  this  point  in  water,  we  have  to  place  in  the  upper 
basin  the  weight  w^ :  and  to  sink  it  to  the  same  point 
in  a  fluid  A  whose  specific  gravity  is  sought,  the 
weight  w^^  is  required.  Denote  the  weight  of  the 
instrument  itself  by  w,  and  the  weights  of  the  dis- 
placed volumes  of  the  respective  fluids  by  w  and  w": 
then  we  have 

w'  ^^  w  -\-  Wi,     and  w"  =  w  -\-  w,,. 

and  hence 

W^^     W  -\-  Wii 

W'  W  -{-  Wj' 

This  hydrometer  may  be  employed  to  determine 
the  specific  gravity  of  solids.  Thus,  the  instrument 
being  placed  in  water  :  1st,  let  it  be  sunk  to  the  fixed 
point  P,  by  j)utting  the  necessary  weight  iv^  in  the 
upper  basin ;  2d,  removing  the  weight,  and  putting 
the  solid  whose  specific  gravity  is  sought  in  its  place, 
let  w^^  be  the  weight  which,  added  to  the  same  basin, 
will  sink  the  instrument  to  P ;  3d,  transferring  the 
solid  to  the  lower  basin,  let  u\^^  be  the  weight  required 
in  the  up23er  basin  to  sink  the  hydrometer  to  the 
same  point.  Employing  w  and  w'  as  above,  and  de- 
noting the  weight  of  the  solid  in  air  by  x,  and  the 
weight  of  the  volume  of  water  displaced  by  the 
solid  by  y,  we  have 

W'    =1    ID  -\~  X  -\-  Wii, 

w'  +  y  —  w-\-x-\-w,„; 


HYDROSTATICS.  227 

and  hence  x  =:  w,  —  w,,^ 

and  y  —  w,„  —  w,„ 

and  ,^^^^^1^2!hL, 

y         wjn—wji 

27.  Gases. 

To  find  the  specific  gravity  of  air,  or  any  other 
gas,  referred  to  water  as  the  standard,  let  a  glass 
flask  be  weighed,  1st,  when  filled  with  the  gas ;  2d, 
when  filled  with  water ;  and,  3d,  when  empty  :  then 
denoting  the  respective  weights  by  w,  w  and  w\ 
and  the  specific  gravity  required  by  5,  we  have 


from  which  the  specific  gravity  referred  to  air  can 
easily  be  found,  that  of  air  referred  to  water  being 
supposed  known. 

In  the  preceding  articles,  no  allowance  has  been 
made  for  the  effect  of  the  atmosphere,  in  virtue  of 
which  the  body,  when  weighed  in  it,  is  urged  verti- 
cally upwards  with  a  force  equal  to  the  weight  of 
the  air  which  it  displaces ;  nor  for  the  deviations 
from  the  standard  temperature  and  pressure.  These 
reductions  belong  to  physics. 


PRESSURE    OF    THE    ATMOSPHERE.       ELASTICITY    OF 
COMPRESSIBLE   FLUIDS.       THE    BAROMETER. 

28.  If  a  glass  tube,  about  three  feet  in  length, 
closed  at  one  end,  be  filled  with  mercury,  and  then 
be  inverted  in  a  vessel  containing  the  same  fluid,  the 
mercury  will  still  occupy  a  certain  portion  FH  [Fig. 


228  MECHANICS. 

20]  of  the  tube  above  the  level  of  the  fluid  in  the 
vessel.  If  we  denote  the  area  of  the  section  GF  by 
1,  the  vertical  height  FH  of  the  column  of  mercury 
by  A,  and  the  density  of  the  mercury  by  d,  the 
pressure  exerted  by  the  column  upon  the  stratum  GF 
will  be  expressed  by  the  product  hdg ;  and  denoting 
this  pressure  by  n,  we  shall  have 

n  =  hdg. 

This  pressure  is  evidently  in  equilibrium  with  the 
pressure  of  the  atmosphere  upon  the  free  surface  of 
the  mercury  in  the  vessel.  Thus  the  weight  of  the 
column  of  mercury  is  equal  to  the  weight  of  a  verti- 
cal column  of  air  of  the  same  base,  extending  to  the 
limits  of  the  atmosphere.  The  glass  tube,  employed 
as  just  described  is  the  essential  part  of  the  instru- 
ment called  the  barometer.  At  the  same  place  on 
the  earth's  surface,  the  height  h  of  the  barometric 
column,  which  thus  measures  the  pressure  of  the 
atmosphere,  is  subject  to  small  variations,  both  pe- 
riodic and  irregular.  Its  mean  value,  at  the  level  of 
the  sea,  is  about  thirty  inches.  From  this  it  can 
readily  be  shown  that  all  bodies,  at  the  level  of  the 
sea,  are  subjected  to  a  pressure  of  nearly  fifteen 
pounds  to  the  square  inch.  The  height  of  the  column 
of  air  which  is  in  equilibrium  with  the  column  of 
mercury,  on  the  hypothesis  that  the  air  is  at  the  same 
density  throughout  the  column  as  at  the  surface  of  the 
earth,  is  easily  found.  For  the  weight  of  the  column 
of  mercury  is  expressed  by  hdg;  and  denoting  the 
height  of  the  column  of  air  by  x,  and  the  density 


HYDROSTATICS.  229 

of  the  air  by  D,  the  weight  of  the  column  of  air  is 
expressed  by  xD^ ;  and  hence  we  have 
xDg  =z  hdg, 

and  X  ^=i  —h 

=  ^X3«  =  4,99  miles. 

29.  Elastic  force  of  air  or  other  gas,  proportional  to 
its  density. 

It  has  already  been  stated,  in  Art.  5,  that  the 
elastic  force  of  a  compressible  fluid  is  proportional 
to  its  density.  This  principle,  called  from  the  name 
of  its  discoverer,  the  law  of  Mariotte,  is  established  by 
experiment.  Let  ABC  [Fig.  21]  be  a  bent  tube  of 
uniform  bore,  closed  at  C  and  open  at  A,  and  having 
its  branches  A  and  C  vertical.  Let  mercury  be  poured 
into  the  tube,  and  so  adjusted  to  the  air  or  other  gas 
in  the  branch  C  as  to  stand  at  the  same  level  mn  iii 
the  two  branches.  The  air  confined  in  C  will  then 
be  pressed  by  a  force  equal  to  the  weight  of  the 
barometric  column.  Suppose  the  height  of  the  column, 
at  the  time  of  the  experiment,  to  be  just  thirty 
inches  :  then  if  we  double  the  pressure  upon  the  air 
in  C,  by  pouring  mercury  into  the  tube  till  its  height 
pr  in  the  branch  A,  above  its  level  pq  in  the  branch 
C,  becomes  thirty  inches,  the  air  will  be  found  re- 
duced to  one-half  its  original  volume ;  if  we  treble 
the  pressure,  to  one-third;  if  we  quadruple  it,  to 
one-fourth  its  original  volume,  and  so  on.  Thus  the 
volumes  are  inversely  as  the  pressures ;  and  if  we  denote 


230  MECHANICS. 

by  V  and  v'  the  volumes  of  the  same  mass  of  air 
under  the  pressures  p  and  p\  we  shall  have 

V  :    v'    '.'.   p'    '.   p- 

But  denoting  the  densities  corresponding  to  the 
pressures  p  and  p  by  D  and  D\  we  have 

V  :    V'    :i    D     :    D: 

hence  p     :    p'     ::     D      :     D'  ; 

that  is,  the  pressures  are  directly  as  the  dejisities.  But 
the  pressure  is  the  measure  of  the  elastic  force  of  the 
gas  :  hence  the  elastic  force  of  the  gas  is  proportional 
to  its  density,  and  conversely. 

The  temperature  of  the  gas,  it  must  be  observed, 
is  supposed  to  have  remained  the  same  throughout 
the  experiment. 

30.  Change  of  temperature. 

The  volume  of  an  elastic  fluid  varies  with  its  tempe- 
rature. It  has  been  found,  that  if  a  mass  of  air  have 
its  temperature  raised  from  32°  to  212^  of  Fahr.,  it 
will  acquire  an  increment  of  volume  equal  to  ,367 
of  its  original  volum  e ;  and  that,  between  these 
temperatures  at  least,  the  increments  of  volume  are 
proportional  to  the  increments  of  temperature.  The 
increment  of  volume  for  one  degree  is  then  equal 

to  ^  :=  ,00203  z=  a. 

1°  The  volume  v  of  a  given  quantity  of  gas  at  the 
temperature  t  being  given,  to  find  its  volume  v  at  any 
other  temperature  t' ,  we  denote  its  volume  at  32°  by 
X  :  then  we  have 


HYDROSTATICS.  231 

V    ^^  X  -\-  {t  —  32)a2;, 
and  v'  =  x-\-{t'  —  32)aa:, 

and  hence  -  =    Y^jr=--Wa' 

and  effecting  the  division  indicated  in  the  second 
member,  and  retaining  only  the  terms  involving  the 
first  power  of  a,  we  have 

V'  z=z  {l-{-{t'  —  t)a)v. 

2"  To  find  an  expression  for  the  pressure  or  elastic 
force  in  terms  of  the  density  and  temperature,  let  v 
denote  the  volume  of  a  mass  of  air  at  32°,  and  v' 
its  volume  at  t,  under  the  atmospheric  pressure  n  : 
then  by  the  above  we  have 

v'  =  v-\-{t  —  32)  ,00203  V 
—   [l-\-{t  —  ^2)a}v. 
But   denoting  the   densities  corresponding  to  the 
volumes  v  and  v'  by  D  and  D',  we  have 

V    :    V'    ::    D'    :    D, 
or  V    :    [i-\-{t  —  B2)a]v    ::    D'    :    D; 

and  hence  D    —  -— — - — ^^^. 

1  +  ff  — 32ja 

Now  let  the  density  and  pressure  be  supposed  to 

change    from  D'    and   n  to   p  and  p   respectivel}^ 

without  any  alteration  in  the  temperature  of  the 

air  :  then 

D'    :    p    ::    u    :   p; 

and  hence  p  =z  7tP[1H-(^ — 32)a] 

=  ^■h-gp[l  +  (t-S2)a]. 


232  MECHANICS. 

31.  Measurement  of  heights  hy  the  barometer. 

The  barometer  may  be  employed  to  determine 
the  difference  of  level  between  two  stations ;  as,  for 
example,  betAveen  the  base  and  top  of  a  mountain. 

To  investigate  a  formula  for  this  purpose,  let  AB 
[Fig.  22]  represent  a  vertical  cylindrical  column  of 
air,  extending  from  A  at  the  level  of  the  sea,  indefi- 
nitely upwards,  and,  to  aid  the  conception,  supposed 
to  be  separated  from  the  surrounding  atmosphere 
by  an  immovable  envelope.  Conceive  this  column 
divided  into  horizontal  strata  of  equal  thicknesses, 
the  thickness  being  so  small  that  the  density  of  the 
air  in  each  stratum  may  be  considered  the  same 
throughout  the  stratum.  Let  the  common  thickness 
of  the  strata  be  denoted  by  x  :  then  the  heights  of 
the  upper  surfaces  of  the  strata,  commencing  at  the 

lowest  stratum,  will  be  denoted  by  x,  2x,  3j:, 

nx  =  X .  Let  the  heidits  of  the  barometer  corre- 
spending  to  the  successive  strata  be  denoted  by  A, 

h',  h", A  ;  and  the  densities  of  the  air  in  these 

strata,  by  D,  D ,  D",  etc.,  that  of  mercury  being 
taken  for  the  unit ;  and  let  the  temperature  of  the 
whole  column  be  supposed  to  be  32^  of  Fahr.  Also 
let  the  intensity  of  gravity  be  denoted  by  1,  and 
let  it  be  supposed  constant  throughout  the  column. 
Assuming  the  horizontal  base  of  the  column  to  be 
the  unit  of  area,  the  weight  of  the  first  stratum  of 
air  will  be  expressed  by  xD ;  the  weight  of  the 
column  of  mercury  whose  base  is  also  the  unit  of 
area,  and  whose  height  is  h,  will  be  expressed  by  h; 


HYDROSTATICS.  233 

and  the  weight  of  the  column  of  mercury  whose 
height  is  A',  by  A';  and  we  shall  have 

xD  =z  h  —  h'. 
Since  the  density  of  the  air  at  any  point  is  propor- 
tional to  the  pressure,  that  is,  to  the  height  of  the 
barometer  at  that  point,  if  we  denote  the  constant 
ratio  of  A  to  D  by  c,  we  shall  have 

n  =  ch; 
and  hence  cxh  z=.  li  —  h\ 

and  h'  =^  h  —  cxh 

—  h{l  —  cx). 

In  like  manner,  we  get  for  the  second  stratum 

xD>  —  h'  —  h"; 
and  since  D'  z=.  ch', 

cxh'  =  h'  —  h"  ; 
and  hence  h"  —  h'{l  —  ex) 

=  Ji{l  —  cxf; 

and  generally  for  the  nth.  stratum, 

h,  =  h{l  —  cxy\ 

Taking  the  Naperian  logarithm  of  each  member  of 
this  equation,  we  have 

Log  hj  —  Log  h 


Logfl- 

cx) 

1 
Loga- 

-ex) 

(Log  h  — 

-Log/0; 

X, 
X    ' 

1 

Tno-Zl 

r-rl 

.(Log/i- 

-Log  A,). 

or,  since  n  =. 

^  — 

x 

But  by  a  well  known  logarithmic  series,  we  have 


234  MECHANICS. 

Log(l — ex)  =1  — ex  —  ^(^'x^  —  etc.; 

or,  neglecting  the  powers  of  x  above  the  first, 

Log  (1  —  ex)  z=  — ex  : 

hence  hy  substitution  we  get 

^/  =  -(Log  A  — Log  A,). 

Denoting  by  M  the  reciprocal  of  the  modulus  of  the 
common  system ;  and  by  *  log'  a  common  logarithm, 
Ave  have  ^ 

logA  =  ^     and  logh,  =  ^; 

and  hence  we  get 

For  any  other  height  x^^,  at  which  the  height  of  the 
barometer  is  A  ^,  we  have 

x„  =  --(log  A  — log  A,/); 

and  hence  we  get 

M 

^,,  —  ^i  =  ^"  =  —  (logA,  — logAJ; 

and  hence,  substituting  the  value  of  c, 

^"  =  ^  (log/* -log  A.). 

If  now  the  temperature  be  supposed  to  change 
from  32°,  the  assumed  temperature,  to  f  any  tempe- 
rature whatever,  the  air  of  the  column  will  either 
expand  or  contract.  If,  for  example,  we  suppose  it 
to  expand,  then  the  points  at  which  the  barometers 
indicate  the  same  pressure  as  before  will  be  more 
elevated,  and  the  distance  z"  between  these  points 


HYDROSTATICS.  235 

will  be  increased.  But  since  air  under  a  constant 
pressure  expands  the  ,00203th  of  its  volume  at  32°, 
for  a  change  of  temperature  of  one  degree,  if  we 
denote  the  volume  of  the  column  at  32°  by  1,  its 
volume  at  f  will  be  1  +  (^  -  32)  ,00203  ;  and  de- 
noting the  new  value  of  the  height  of  the  column 
by  z\  and  recollecting  that  the  heights  are  directly 
as  the  volumes,  we  shall  have 

z"    :    z'    ::    \    :    1  + (^  —  32)  ,00203, 
and  z'  =  \\-\-  (^  —  32)  ,00203]z" 

=  ^  [1  +  (^  -  32)  ,00203]  (log  h,  -  log  K). 

We  have  denoted  the  intensity  of  gravity  by  1 ; 
but  it  has  been  found  by  experiments  with  the  pen- 
dulum, that  if  the  intensity  of  gravity  on  the  parallel 
of  45°  be  denoted  by  1,  its  intensity  at  any  other 
latitude  V^  will  be  expressed  by  (1  —  ,002837  cos  2ip); 
and  if,  for  example,  we  suppose  2  V^  to  be  greater  than 
90°,  the  extremities  of  the  column  will  be  depressed, 
and  its  volume,  and  consequently  its  length,  will  be 
diminished. 

Since  the  volumes  are  inversely  as  the  pressures 

and  directly  as  the  heights,  and  hence  the  heights 

inversely  as  the  pressures,  if  we  denote  the  new 

value  of  the  height  of  the  column  by  z,  we  shall  have 

z    :    z'    ::    1    :    (1  — ,002837 cos 2i/)), 

and  z 


fl  — ,002837  cos  2V;' 


Mh  [1  +  ff  — 32;,00203]  n      7,       ^      J.  \  f*\ 

^  =  -^  a-' 002837  cos  2^;    (log  A.  -  log  A.) W 


236  MECHANICS. 

We  have  supposed  the  air  to  be  at  the  same  tempe- 
rature throughout  the  column ;  but  this  is  never  the 
case.  We  take  for  t  the  mean  of  the  temperatures 
t'  and  t"  at  the  extremities  of  the  column,  and  thus 
have 

A  correction  must  be  applied  for  the  effect  of  the 
unequal  temperatures  at  the  extremities  of  the 
column,  upon  the  heights  of  the  barometers.  The 
temperature  of  the  mercur}^  in  the  barometer  (which 
is  determined  by  a  very  delicate  thermometer  at- 
tached to  the  instrument),  will  in  general  differ  from 
that  of  the  air.  We  will  denote  the  temperatures  of 
the  mercury  in  the  lower  and  upper  barometers  re- 
spectively by  t'"  and  f"" :  then  since  mercury  expands 
about  ,0001th  of  its  volume  for  each  degree,  the 
observed  height  h^^  must  be  increased  by 

and  in  place  of  A^,,  we  must  employ 

h„  =  [1  +  {t'"-  tn  ,0001]h„. 
To  find  the  value  of  the  co-efficient  ^^,  we  pro- 
ceed as  follows  :  In  some  particular  case,  the  quantity 
z  (the  height  of  a  lofty  mountain,  for  example)  must 
be  measured  trigonometrically  with  great  accuracy  ; 
and  the  corresponding  values  of  h^  and  A  ^,  t'  and  t", 
t"'  and  r,  and  i^,  must  be  determined  by  observation. 
We  shall  then  have  the  numerical  values  of  all  the 
quantities  contained  in  the  equation  {t),  except  this 


HYDROSTATICS.  237 

coefficient;  and  hence  it  can  be  found.  Its  value, 
obtained  by  this  method,  is  201U^'',6S.  This  coeffi- 
cient can  also  be  obtained  directly  by  determining 

the  ratio  ^  experimentally.  We  have  then,  as  the 

final  result  of  the  preceding  investigation, 

When  a  more  accurate  formula,  in  which  account 
is  taken  of  the  variation  of  gravity  on  the  same  ver- 
tical, is  employed  in  determining  the  co-efficient  ^ 

by  the  method  first  indicated,  its  value  is  found  to 
be  20052^'*',24 ;  a  result  which  agrees  almost  exactly 
with  that  obtained  directly  by  the  second  method, 
it  being  20050^'^',  18.  The  difference  between  the  re- 
sults found  by  the  two  formulas  is  the  compensation 
due  to  the  neglect,  in  the  preceding  investigation, 
of  the  variation  of  gravity  on  the  same  vertical. 


PART  FOURTH. 


HYDRODYNAMICS. 


1.  Efflux  of  fluids. 

Let  ABCD  [Fig.  23]  represent  a  vessel  having  in 
its  bottom  a  very  small  orifice  GH,  and  filled  with  a 
fluid  up  to  the  level  AB.  Also  let  GHIK  represent  the 
column  of  fluid  directly  above  the  aperture  GH,  and 
EFHG  an  indefinitely  thin  stratum  of  the  column 
immediately  contiguous  to  GH.  Let  EG,  the  height 
of  the  stratum,  be  denoted  by  h'  :  then  if  we  suppose 
the  accelerating  force  which  acts  upon  the  stratum 
to  be  gravity  alone,  the  velocity  v'  with  which  it  Avill 
leave  the  orifice  will  be  expressed  by  the  equation 

But  the  velocity  with  which  the  stratum  actually 
leaves  the  orifice  is  due  to  its  own  weight,  and  the 
weight  of  the  column  of  fluid  EFIK  directly  above  it. 
Denoting  then  the  actual  accelerating  force  by  g' ,  an  1 
the  corresponding  velocity  by  v,  we  have 


But  denoting  the  height  GK  of  the  entire  column  by 
h,  we  have,  since  the  weights  of  the  columns  EH  and 
KH  are  as  their  heights, 


240  HYDRODYNAMICS. 

g     :     g'     :   :     h'     :     h, 
or  g'k'  =  gh, 

and  hence 

Now  the  second  member  of  this  equation  is  the 
expression  of  the  velocity  which  a  material  point 
would  acquire  in  falling  through  the  space  h.  Hence 
it  appears  that  the  velocity  with  ivhich  a  fluid  issues 
froin  a  small  orifice  in  the  bottom  of  a  vessel,  is  equal  to 
the  velocity  which  a  heavy  body  would  acquire  in  falling 
through  a  space  equal  to  the  height  of  the  level  of  the  fluid 
above  the  orifice. 

We  have  supposed  the  orifice  to  be  in  the  bottom 
of  the  vessel,  and  horizontal;  but  since  fluids  press 
equally  in  all  directions,  the  above  reasoning  is  appli- 
cable, whatever  the  inclination  of  the  plane  of  the 
orifice ;  the  only  condition  being  that  the  dimensions 
of  the  orifice  shall  be  small,  compared  with  the  dimen- 
sions of  the  vessel,  and  also  with  the  distance  of  the 
orifice  below  the  surface  of  the  fluid. 

The  principle  just  enunciated  is  therefore  true  when 
the  orifice  is  in  any  face  of  the  vessel,  and  whether 
the  face  be  horizontal,  vertical,  or  oblique. 

This  principle  may  be  verified  by  experiment. 
Thus  if  a  vessel  ABDC  [Fig.  24],  having  a  very  small 
orifice  G  in  the  horizontal  face  FE,  be  filled  Avith 
water  up  to  the  level  AB,  the  vertical  height  GH  of 
the  jet  of  fluid  which  will  issue  from  G  will  be  found 
to  be  very  nearly  equal  to  BF  the  height  of  the  level 
of  the  fluid  above  the  orifice ;  and  we  may  assume 


HYDRODYNAMICS.  241 

that  it  would  be  exactly  equal  to  it,  were  there  no 
disturbing  force  to  diminish  GH. 

2.  Head  of  water. 

The  height  h  is  called  the  head  of  water.  For  any 
two  heads  h  and  h' ,  for  whicli  the  corresponding  velo- 
cities are  v  and  v' ,  we  have 

v  =  \^2gh,  and  v'  =  \^2gh', 

and  hence 

V     :     v'     :   :     \/h     :     s/h' ; 

M^hence  it  appears  that  for  different  heads  of  water, 
other  circumstances  being  the  same,  the  velocities  are 
to  each  other  as  the  square  roots  of  the  heads :  a  princi- 
ple which  has  also  been  verified  by  experiment. 

3.  In  the  foregoing,  it  has  been  assumed  that  the 
exterior  pressures  at  the  orifice,  and  at  the  upper  sur- 
face of  the  fluid  are  equal.  When  an  additional  pres- 
sure is  applied  to  the  latter,  the  efiect  is  obviously 
the  same  as  would  result  from  increasing  the  height 
of  the  fluid ;  and  if  we  denote  by  h'  the  vertical  height 
of  the  column  of  fluid  required  to  furnish  an  equiva- 
lent for  the  additional  pressure,  we  shall  have 

The  quantity  h'  is  determined  by  the  equation 
A'  =  -  ;  in  which  p  is  the  weight  of  a  cubic  foot  of 
the  fluid  contained  in  the  vessel,  P  the  additional 
pressure,  and  s  the  fluid  surface  on  which  it  acts. 

When  the  additional  pressure  is  applied  at  the 
orifice,  the  effect  is  the  reverse  of  that  in  the  case 
just  considered,  and  h'  becomes  negative,  and  we  have 

31 


242  HYDRODYNAMICS. 

V  =  \/2g(/i  —  h'). 

4.  Discharge ;  theoretic  discharge  ;  actual  discharge. 
The  volume  of  fluid  which  passes  through  an  orifice 

in  a  second  of  time,  is  called  the  discharge  of  the 
orifice. 

If  all  the  particles  in  the  vem  of  fluid  be  supposed 
to  have  the  same  velocity  v  =  s/lgh,  and  to  issue  in 
parallel  lines  at  right  angles  to  the  plane  of  the  ori- 
fice, the  discharge  will  be  expressed  by  the  product 

in  which  s  denotes  the  area  of  the  orifice. 

This  is  the  theoretical  discharge :  the  actual  dis- 
charge is  always  found  to  be  less  than  it,  and  is  de- 
rived from  the  theoretical  b}^  multiplying  the  latter 
by  a  certain  coefficient  ??^,  the  value  of  which  is  deter- 
mined by  experiment.  We  thus  have  for  the  actual 
discharge, 

and  denoting  by  Q  the  quantity  of  fluid  discharged 
in  t  seconds,  the  height  h  being  supposed  constant, 
we  get 

Q  =  tms  \^2gh. 

5.  Contraction  of  the  fluid  vein. 

The  difference  between  the  theoretical  and  actual 
discharge  is  due  to  a  contraction  which  takes  place 
in  the  vein  of  fluid  as  it  issues  from  the  orifice.  The 
vein  thus  contracted  is  represented  in  figure  25,  in 
which  AB  represents  the  diameter  of  the  orifice,  sup- 
posed to  be  circular ;  MN,  the  diameter  of  the  section 


I 


HYDRODYNAMICS.  243 

of  greatest  contraction ;  and  CD,  its  distance  from  the 
orifice.  These  dimensions,  AB,  MN  and  CD,  have 
been  found  to  be  to  each  other  as  the  numbers  1,000, 
0,787,  0,498,  or  nearly  as  10,  8,  5. 

6.  Determination  of  the  coefficient  m. 

To  determine  the  vahie  of  m,  denoting  the  areas  of 
the  orifice  and  the  section  of  greatest  contraction  by 
s  and  5'  respectively,  we  have 

5     :     s'     :  :     r     :     0,787^; 

and  hence 

s'  =  0,787^5 

=  0,6195 

=  0,62s  nearly. 

Now  since  we  may  suppose  the  fluid  to  pass  through 
the  section  of  greatest  contraction  with  the  velocity 
due  to  the  head  A,  the  actual  discharge  will  be 

or  0,62s  fi/2gh. 

We  thus  get  m  equal  to  0,62.  Actual  measurements 
of  the  discharge  give  very  nearly  the  same  value 
for  m^. 

7.  Efflux  through  short  tubes. 

If  the  fluid,  instead  of  issuing  directly  from  the  side 
of  the  vessel,  be  discharged  through  a  short  cylindrical 
tube  or  adjutage  exterior  to  the  vessel,  the  length  of 
the  tube  being  only  two  or  three  times  its  diameter,  the 
coefficient  m  will  be  increased  to  0,82.     If  the  tube 

*  The  ratio  of  s  to  s'  is  not  rigorously  independent  of  h,  but  so  nearly 
so  that  it  is  usually  regarded  as  independent  of  it. 


244  HYDRODYNAMICS. 

be  fixed  within  tlie  vessel,  m  will  be  reduced  nearly 
to  0,50. 

In  the  several  cases  just  considered,  the  thickness 
of  the  face  or  side  of  the  vessel  in  which  the  orifice 
exists  is  supposed  to  be  very  small,  compared  Avith 
the  dimensions  of  the  orifice.  An  orifice  in  a  thick 
side  would  obviously  be  equivalent  to  a  short  tube 
inserted  in  an  orifice  in  a  thin  side. 

8.  Effiux  through  large  rectangular  lateral  orifices. 

This  case  is  represented  in  figures  26  and  27 :  in 
the  first  of  which,  the  upper  side  of  the  orifice  coin- 
cides with  the  surface  of  the  water ;  and  in  the  sec- 
ond is  situated  a  little  below  and  parallel  to  it.  In 
the  first  case,  the  aperture  is  called  a  weir. 

1st  Case.  Let  ABIK  represent  the  surface  of  the 
water.  Let  the  w^hole  rectangular  orifice  ABCD, 
which  is  supposed  to  be  vertical,  be  divided  by  hori- 
zontal and  vertical  lines  into  indefinitely  small  rect- 
angles. At  any  one  of  these  elementary  orifices,  as 
P  for  example,  for  which  AP  =  x,  the  fluid  will  issue 
with  a  velocity  equal  to  \^2gx ;  and  denoting  this  velo- 
city by  ?/,  we  shall  have 

y=W^,  and  if  =  1gx. 

This  is  the  equation  of  a  parabola,  whose  parameter 
is  2g.  Hence  if  with  AD  as  an  axis  and  A  as  a  ver- 
tex, we  describe  the  arc  A  MO  of  a  parabola  having 
2g  as  its  parameter,  the  ordinate  PM  of  the  orifice  at 
P  will  represent  the  velocity  Avith  wdiich  the  fluid 
will  issue  from  that  orifice.  The  discharge  from  any 
one  of  the  elementary  orifices  will  then  be  equal  to 


HYDRODYNAMICS.  245 

the  volume  of  a  prism  whose  base  is  the  orifice  and 
height  the  corresponding  ordinate.  Hence  it  is  evi- 
dent that  the  discharge  of  the  whole  rectangle  ABCD 
will  be  equal  to  the  volume  of  the  prism  whose  base 
is  the  parabolic  segment  ADO,  and  height  the  breadth 
AB  of  the  rectangle.  Denoting  the  height  AD 
by  A,  we  have  D0=^V2p;  and  by  a  well  known 
property  of  the  parabola,  the  parabolic  segment 
=  |AD  X  DO  =  p  X  ^2p.  Consequently,  if  the 
breadth  AB  be  denoted  by  /,  the  volume  required,  or 
the  discharge  of  the  orifice,  will  be  expressed  by 

To  find  an  expression  for  the  mean  velocity  (y), 
that  is,  the  velocity  which  must  be  common  to  all 
the  points  of  the  orifice,  in  order  that  the  discharge 
may  be  the  same  as  that  due  to  the  actual  veloci- 
ties, we  have  for  the  expression  of  the  discharge  the 
product  h  .  I  .  v;  and  since  this  is  also  expressed  by 
%h  .  I  .  A^2gh,  we  have 

h  .  I  .  V  =  ^h  .   I  .  \^2gh, 

and  hence 

V  =--  S  ^^2^h. 

2d  Case.  When  the  upper  side  of  the  orifice  is 
below  the  surface  of  the  fluid,  as  in  figure  27,  in  which 
the  orifice  is  represented  by  A'  B'  CD,  the  discharg;^ 
will  evidently  equal  the  discharge  from  an  orifice 
ABCD  diminished  by  the  discharge  from  an  orifice 
ABB' A'.  Denoting  then  the  heads  AA'  and  AD  by 
h'  and  h,  we  have  for  the  required  expression, 


246  HYDRODYNAMICS. 

The  above  expressions  give  the  theoretical  dis- 
charges. To  deduce  from  them  the  actual  discharges, 
we  must  multiply  them  by  certain  numerical  co-effi- 
cients derived  from  experiment. 

9.  Efflux  under  variable  pressure. 

We  shall  consider  only  the  most  simple  case ;  that 
in  which  the  vessel  is  prismatic,  and  the  orifice  in  the 
bottom.  Let  x  denote  the  variable  head,  and  h  its 
greatest  value ;  s" ,  the  area  of  the  transverse  section 
of  the  vessel ;  and  s,  the  area  of  the  orifice.  The 
velocity  v  of  efflux  will  be  expressed  by  \/2gx ;  and 
to  find  the  velocity  v'  with  which  the  upper  surface 
of  the  water  descends,  we  have 


and  hence 


s 


V2gx. 


This  is  the  theoretical  velocity.     If  we  denote  the 
actual  velocity  by  v"  ^  we  shall  have 


Here  the  initial  velocity  is  y  2  l~j  gk,  and  the 

final  velocity  is  zero,  and  v' '  varies  as  the  a^x  ;  that 
is,  as  the  square  root  of  the  space  to  be  described. 
Hence  this  case  is  the  same  as  that  of  a  body  pro- 
jected vertically  upwards  [Art.  9,  Dynamics.]^     But 

*  We  have  (Art.  9,  Dynamics) 

v  =  a  —  gt,     or  t  = 


g 
and  s=at  —  ^gf^; 


HYDRODYNAMICS.  247 


in  that  case,  the  time  required  to  reduce  the  velocity 
to  zero  is  equal  to  the  initial  velocity  divided  by  the 
accelerating  force ;  and  here  the  accelerating  force  is 

y-fTJ  g'  Hence,  denoting  the  time  in  which  the  ves- 
sel will  empty  itself  by  t,  we  have 


\^'2gh 


/msV 


'  ms  ^  V2gh 
__  2s"h 

T' 
where  q  denotes  the  discharge  due  to  the  head  h. 

But  if  we  denote  by  t'  the  time  in  which  a  volume 
of  liquid  =s"h  =  the  volume  of  the  vessel  would  be 
discharged  under  the  constant  head  h,  we  get 

from  which  we  get 

the  time  of  ascent  =  —  , 

S 

and  the  greatest  elevation  =  — . 

2g 

Denotino'  by  s'  the  space  which  remains  to  be  described  at  the 
end  of  t  seconds,  we  have 

or  s'  =  ^-at-^igt\ 


or 


2g^ 

and  V  =  \/2gs' ; 

that  is,  the  velocity  as  the  square  root  of  the  space  to  be  described. 


248  HYDRODYNAMICS. 

t'q  =-  s"h 

and  2t'  =  '^, 

q 

and  hence  t  =  2t'. 

Whence  it  appears  that  the  time  in  which  a  prismatic 

vessel  empties  itself,  is  double  the  time  required  for  the 

efflux  of  the  same  quantity  of  fluid  in  the  case  in  tvhich 

the  head  remains  the  same  as  at  the  commencement  of  the 

efflux. 


INSTRUMENTS  AND  MACHINES. 


10.   The  siphon. 

Let  ACB  [Fig.  28]  represent  an  inverted  curved 
tube,  of  which  the  part  ECD  is  filled  with  a  liquid. 
The  surface  of  the  liquid  at  E  is  urged  upwards  by 
the  atmospheric  pressure,  and  downwards  by  the  pres- 
sure of  the  liquid  in  the  branch  AC.  Hence,  denot- 
ing the  height  of  a  column  of  the  liquid  employed, 
whose  weight  is  equal  to  the  pressure  of  the  atmos- 
phere, by  h',  the  vertical  height  CF  of  the  liquid  in 
the  branch  AC,  by  A'  ;  the  density  of  the  liquid  by  d, 
and  the  intensity  of  gravity  by  g,  we  have  for  the 
total  pressure  at  E,  the  expression 

gdh  —  gdh'. 

Also  denoting  CG  by  A",  we  have  for  the  total  pres- 
sure at  D,  the  expression 

gdh  —  gdh". 

Then  if  A' '  is  less  than  A,  and  A'  is  greater  than  A' ' , 
the  total  pressure  at  D  will  be  greater  than  the  total 
pressure  at  E,  and  the  liquid  Avill  flow  from  the  tube 
in  the  direction  DCE. 

Now  let  ACB  [Fig.  29]  be  a  curved  tube  immersed 
in  a  fluid  as  represented  in  the  figure,  and  suppose  the 
air  to  be  withdrawn  from  the  tube ;  then  it  follows 
from  the  foregoing,  that  the  water  will  flow  from  the 


250  INSTRUMENTS    AND    MACHINES. 

vessel  through  the  tube,  and  will  continue  to  do  so  as 
long  as  CG,  which  increases  as  the  surface  of  the 
water  falls,  does  not  exceed  the  limit  h. 
11.  Pumps. 

A  pump  is  a  machine  for  raising  water  by  means 
of  atmospheric  pressure. 
The  sucking  pump. 

The  sucking  pump  consists  of  two  cylindrical  tubes 
CDF'E',  GHRM  [Fig.  30],  accurately  fitted  to  each 
other ;  the  one  called  the  body  of  the  pump,  the  other 
the  sucking  pipe.  To  the  former  is  adapted  an  air- 
tight piston  AB,  having  an  aperture  at  its  centre, 
closed  by  a  valve  S,  called  the  piston  valve,  opening 
upwards.  The  piston  is  movable  through  a  certain 
space  EE',  called  the  play  of  the  piston.  At  CD, 
where  the  cylinders  are  connected,  the  passage  is 
closed  by  a  second  valve  S' ,  called  the  sleeping  valve, 
also  opening  upwards.  At  a  point  0,  a  little  above  E' , 
is  a  lateral  orifice  at  which  the  water  is  discharged. 

To  explain  the  operation  of  the  pump,  let  MR  be 
the  level  of  the  water  in  the  reservoir.  Let  the  pis- 
ton be  at  EF  the  lower  limit  of  its  play,  both  valves 
being  closed,  and  the  air  within  the  pump  of  the  same 
elasticity  as  the  external  air.  Then  when  the  piston 
is  raised  to  the  higher  limit  E'F',  the  valve  S'  will 
open,  because  of  the  greater  elastic  force  of  the  air 
in  the  sucking  pipe ;  and  a  portion  of  this  air  will 
enter  the  body  of  the  pump.  The  elastic  force  of 
the  air  in  the  whole  interior  of  the  pump  below  E'F' 
being  thus  diminished  by  expansion,  the  water  will 


INSTRUMENTS   AND    MACHINES.  251 

rise  in  the  sucking  pipe  to  some  point  R' ,  such  that 
the  pressure  due  to  the  water  raised,  together  vv  ith 
the  elastic  force  of  the  air,  shall  be  in  equilibrium 
with  the  pressure  of  the  atmosphere.  The  valve  S' , 
being  then  equally  pressed  on  both  sides,  will  close 
by  its  own  weight.  If  now  the  piston  be  depressed, 
the  valve  S  will  open,  and  a  portion  of  the  air  in  the 
body  of  the  pump  will  escape ;  and  when  the  piston 
reaches  its  former  position  EF,  the  valve  will  close, 
leaving  the  air  in  the  space  EFDC  of  the  same  elas- 
ticity as  the  external  air.  Each  elevation  and  de- 
pression, or  double  stroke  of  the  piston,  will  produce 
a  similar  effect ;  and  after  a  certain  number  of  repe- 
titions, the  water  will  enter  the  body  of  the  pump, 
and,  passing  through  the  valve  S,  will  reach  the 
orifice  at  0.  Then  at  each  depression  of  the  piston, 
a  volume  of  water,  equal  to  the  cylinder  whose  base 
is  the  horizontal  section  of  the  body  of  the  pump, 
and  whose  altitude  is  the  play  of  the  piston,  will  pass 
through  the  valve  S,  and,  by  the  succeeding  eleva- 
tion, be  raised  to  the  orifice.  In  order  that  the  water 
may  enter  the  body  of  the  pump,  MG  must  evidently 
be  less  than  the  height  h  of  the  vertical  column  of 
water  whose  weight  is  equal  to  the  atmospheric  pres- 
sure ;  and  in  order  that  the  maximum  volume  may 
be  delivered  at  the  orifice,  Me'  must  not  exceed  this 
limit.  If  we  have  Me  <  A,  but  Me'  >  A,  a  greater  or 
less  volume  will  be  raised  to  0,  according  to  the  ratio 
of  Me'  to  h. 

12.  Deteimination  of  the  height  of  the  water,  and.  the 


252  INSTRUMENTS   AND    MACHINES. 

elastic  force  of  the  air  in  the  pump,  after  any  number  of 
strokes  of  the  piston. 

1st  Case.  Before  the  water  has  reached  the  sleeping 
valve. 

Let  N  be  the  point  reached  by  the  water  after  any 
number  of  strokes  of  the  piston,  the  piston  being  at 
the  lower  limit  EF.  Let  HR  and  HN  be  denoted  by 
a  and  y,  the  cylindrical  volumes  CDFE,  CDF'E', 
by  V  and  v'  ;  the  horizontal  section  of  the  sucking 
pij^e,  by  s ;  the  height  of  the  column  of  water  whose 
weight  is  equal  to  the  atmospheric  pressure,  by  h ; 
and  the  height  of  the  column  which  would  produce 
the  same  pressure  as  the  air  contained  in  the  suck- 
ing pipe,  by  x.  Then  the  height  of  the  water  in  the 
sucking  pipe  will  be  a  —  ?/,  and  will  be  such  that  the 
weight  of  the  column  of  liquid  MN,  together  with  the 
elastic  force  of  the  air  in  the  space  NOGH,  will  be 
equal  to  the  atmospheric  pressure.  Hence,  denoting 
the  density  of  the  liquid  by  d,  and  the  intensity  of 
gravity  by  g,  we  have 

gdx  -^  gd{a  —  y)=  gdh  ; 

or,  putting  h  —  a  =  b, 

x  —  y  =  b [1] 

Now  the  volume  of  air  NOGH,  whose  elastic  force  is 
gdxy  is  equal  to  sy,  hence,  since  the  volumes  are 
inversely  as  the  pressures,  its  volume  at  the  elastic 

force  gdh  will  be  ^^^ ;  and  thus  the  total  volume  of 
air  in  the  interior  of  the  pump,  viz :  the  volume 
NOGH  +  the  volume  CDFE,  at  the  elastic  force  gdh, 


INSTRUMENTS    AND    MACHINES.  253 

will  he  V  -\-^.     If  now  the  pi.ston  be  raised  to  E'F', 

the  water  will  rise  to  some  point  N',  and  the  air  will 
become  of  the  same  elastic  force  throughout.  Let 
x'  and  y'  denote  the  new  values  assumed  by  x  and  y : 
then  we  shall  have,  as  above, 

and  x'  —  y'  =h [2] 

The  interior  volume  of  air  is  now  equal  to  v'  -f-  sy' , 
and  its  elastic  force  is  gdx' ,  and  the  volume  of  this 
same  mass  of  air  at  the  elastic   force  gdh  we  have 

found  to  be  ^  -j-  ^.     Hence  we  have 

v'  -\-  sy'     :     V  -{-  -^     :   :     k     :     x', 

and  ;z,,-f  ,X2/  .g. 

From  equations  [1],  [2]  and  [3],  we  get  by  elimina- 
tion, 

,-=_^'±^j(^-^y+^+.=-*.j [5] 

Now  suppose  the  elevation  of  the  w^ater  in  the  suck- 
ing pipe  to  be  zero :  then 

y  =  a,  and  x=  h  ; 

and  substituting  this  value  of  x  in  equations  [4J  auu 
[5],  we  have  the  values  of  x'  and  y' ,  and  hence  of 
a  —  y'  in  known  terms;  that  is,  the  elastic  force  of 
the  air  in  the  sucking  pipe,  and  the  height  of  the 
water  after  the  first  stroke  of  the  piston. 


254  INSTRUMENTS    AND    MACHINES. 

Again,  substituting  in  these  equations,  in  place  of 
X,  the  value  of  x'  just  found,  we  get  the  values  of 
these  same  quantities  after  the  second  stroke  of  the 
piston ;  and  proceeding  thus,  we  can  find  their  values 
after  any  number  of  strokes. 

To  apply  the  preceding  formulae  to  the  case  in 
which  the  water,  after  having  risen  to  a  certain  height 
a  —  y,  ceases  to  rise  any  further,  we  have  only  to 
suppose 

we  thus  get  from  equations  [1]  and  [2], 
and  from  equation  [3], 

x=  — -A. 

To  find  the  height  at  which  the  water  ceases  to 
rise,  we  get  from  equation  [1], 

a  —  y  =  h  —  X 

v' 

From  this  equation,  it  appears  that  if  ^  =  0,  that 
is,  if  the  pump  is  so  constructed  that  the  piston,  at 
the  lower  limit  of  its  play,  touches  the  sleeping  valve, 
the  water  will  rise  to  the  height  h,  the  greatest  height 
possible. 

To  interpret  the  equation  x  =  -r^>  let  the  distances 
F'D  and  F'F  be  denoted  by  /  and  /'  respectively: 
DF  will  then  be  equal  to  /  —  /' ,  and  we  shall  have 


I 


INSTRUMEJ;[^TS    AND    MACHINES.  255 


y  __L  —  /' 
and  hence  ^  ^  ^~^'k 


Now  after  the  stroke  of  the  piston  which  leaves  the 
water  at  the  height  (1_  J,)/,,  the  piston  being  sup- 
posed at  the  lower  limit,  the  elastic  force  of  the  air 
in  the  space  CDFE  will  be  equal  to  gdh ;  and  when 
the  same  mass  of  air  is  expanded  by  the  elevation 
of  the  piston  so  as  to  fill  the  space  CDF'E',  its  elastic 
force  will  be  ^~^^gdh.     Also  gdx  is  the  elastice  force 
of  the  air  in  the  sucking  pipe.     This  equation,  then, 
expresses  the  equality  of  the  pressures  upon  the  oppo- 
site  sides  of  the  sleeping  valve,  in  which  case  the 
valve  will  not  open,  and  consequently  the  elevation 
of  the  water  must  cease.     The  condition  upon  which 
the  further  rise  of  the  water  depends  is  evidently 

2d  Case.     Whm  the  water  is  already  above  the  sleep- 
ing valve. 

Let  us  now  suppose  the  water  to  have  risen  to  some 
point  N' '  above  the  sleeping  valve.  Employing  the 
same  notation  as  in  the  preceding  case,  with  the  ex- 
ception that  s  now  denotes  the  horizontal  section  of 
the  body  of  the  pump,  we  have  v  —  sy  =  the  volume 
of  air  between  the  surfiice  of  the  water  N"  0"  and 
EF  at  the  elastic  force  gdh.  Also  supposing  the 
piston  to  be  raised  to  E'F',  we  have  v'  — sy'  =  the 
volume  of  air  between  the  surface  of  the  water  N' ' ' 


256  INSTRUMENTS    AND    MACHINES. 

0'"  and  E'F',  at  the  elastic  force  gdx' .  Hence  we 
have 

V  —  sy     :     v'  —  sy'     :  :     xf     :     h, 

and  x'  =  ''~'y,h [6] 

We  also  have         x'  +  a-^y'  =  h, 

and  X'  -{-y'  =  h [7] 

From  these  equations  we  get 

To  find  whether  the  water  will  cease  to  rise  before 
it  has  reached  the  limit  A,  we  make  y'  =^y  in  equa- 
tions [7]  and  [6],  and  thus  get 

x'  =  b  —  y,  and  b  —  v  =  — v 

'^'  ^        V  — sy 

Reducing  the  last  equation,  we  have 

sy-  -{-  {{Ji  —  b)  s  —  v')y  =  hv  —  hv' ; 

or,  substituting  h  —  a  for  h, 

^  v'  —  as  +  ,  /  J  {v'-\-asf  —  4:hs{v'  —  v)  \ 
y         2s  V   (  4s^  $  • 

The  water  will  cease  to  rise  when  the  values  of  y  are 
real :  there  may  then  be  two  points  at  which  it  will 
cease  rising.  The  condition  to  be  fulfilled,  in  order 
that  the  water  shall  not  cease  to  rise,  is  obviously 

[v'  -\- asy  <:^Ahs{v'  —  v), 

or  ('-4^')<4A^'^7''\ 


INSTRUMENTS    AND    MACHINES.  257 

(f+«)<4;,p-7-^); 

that  is,  {Me'Y  <  4A  +  FF'. 

13.  Pressure  upon  the  piston 

To  find  the  pressure  to  which  the  piston  is  sub- 
jected when  it  is  ascending  and  the  sleeping  valve 
open,  first  suppose  the  water  not  yet  to  have  reached 
the  piston,  and  denote  the  elasticity  of  the  air  be- 
tween it  and  the  piston  by  gdx ;  also  let  the  height 
of  the  water  in  the  pump  be  denoted  by  h' ;  then 
gdx  =  gdh  —  gdh'  =  the  pressure  upon  the  lower  sur- 
face of  the  piston,  and  gdh  =  the  pressure  upon  the 
upper  surface.  Hence  the  resultant  of  these  oppo- 
site pressures,  directed  downwards,  and  referred  to 
the  unit  of  surface  =  gdh  —  {gdh  —  gdh')  =  gdh' . 

Secondly,  suppose  the  water  to  have  passed  through 
the  piston  valve,  and  denote  the  height  of  the  column 
of  water  which  rests  on  the  piston  hy  h"  ;  also  denote 
the  height  of  the  piston  above  the  level  of  the  water 
in  the  reservoir  by  h' ,  and  suppose  h'  <C  h:  then 
gdh  —  gdh'  =  the  pressure  on  the  lower  side  of  the 
piston,  and  gdh  +  gdh' '  =  the  pressure  upon  the  upper 
side.     Hence  their  resultant 

=  gdh  +  gdh"  —  {gdh  —  gdh')  =  gdh'  +  gdh". 

If  we  have  h'  >  h,  there  will  be  a  vacuum  below 
the  piston,  and  the  resultant  pressure,  directed  down- 
ward, will  be  =  gdh  +  gdh".  Thus  the  pressure 
upon  the  piston,  during  its  ascent,  is  in  general  equal 

to  the  loeight  of  a  column  of  water  whose  base  is  equal 

33 


258  INSTRUMENTS    AND    MACHINES. 

to  the  horizontal  section  of  the  body  of  the  pump,  and 
whose  altitude  is  the  height  of  the  water  in  the  pump. 

14.  Forcing  pump. 

When  the  height  to  which  the  water  is  to  be  raised 
is  greater  than  the  limit  h,  a  modification  of  the  suck- 
ing pump  is  employed,  called  the  forcing  pump.  In 
this  pump,  the  piston  AB  [Fig.  31]  is  not  perforated ; 
and  at  a  point  Q,  a  little  above  CD  and  just  below 
the  lower  limit  of  the  play  of  the  piston,  is  an  orifice, 
to  which  is  adapted  a  tube  PT,  furnished  with  a  valve 
S"  opening  upwards.  The  water  rises  in  the  body 
of  the  pump  in  the  same  manner  as  in  the  sucking 
pump,  following  the  piston  in  its  ascent  to  the  upper 
limit :  then  when  the  piston  descends,  the  valve  S' ' 
opens,  and  a  volume  of  water,  equal  to  the  cylinder 
described  by  the  piston,  is  forced  up  the  tube  PT. 
The  valve  S' '  then  closes.  Each  stroke  of  the  piston 
producing  a  like  effect,  water  may  thus  be  raised  to 
any  desired  height. 

15.  Bramah's  press. 

Let  M  and  N  [Fig.  32]  be  two  hollow  cylinders  con- 
nected by  a  tube  IK,  and  furnished  with  pistons  A 
and  B.  Let  S  and  S'  be  two  valves  opening  as  rep- 
resented in  the  figure ;  and  let  N,  the  smaller  cylin- 
der, terminate  in  a  reservoir  EG  of  water.  When 
the  j)iston  B  is  raised  by  means  of  the  lever  QF,  the 
valve  S  opens,  and  the  water  rises  as  in  the  common 
pump ;  and  when  it  is  depressed,  S  closes,  S'  opens, 
and  the  water  is  forced  into  the  cylinder  M.  When 
the  part  of  the  cylinder  below  A  is  filled  with  water. 


INSTRUMENTS   AND    MACHINES.  259 

each  depression  of  the  piston  B  must  cause  A  to  rise. 
To  determine  the  pressure  thus  produced  upon  the 
base  of  A,  let  the  horizontal  sections  of  the  cylinders 
M  and  N  be  denoted  by  s  and  s'  respectively ;  the 
force  applied  at  Q  by  P ;  and  the  distances  from  the 
fulcrum  F  to  Q  and  0,  by  /  and  /' :  then  the  pressure 
exerted  at  0  will  be  expressed  by 

-I- 

Thus  the  piston  B  exerts  upon  the  water  in  the  cylin- 
der N  a  pressure  equal  to  ^^,  and   this  pressure  is 

transmitted  by  the  liquid  to  the  surface  of  the  piston 
A.  Denoting  then  the  pressure  upon  A  by  R,  we 
have 

s'     :     s     :  :     Pl,     :     R, 

and  R  =  pI^xA-> 

i         s' 

or,  denoting  the  radii  of  the  sections  s  and  s'  by  r 
and  r', 

R  =  I-X~  .P. 

16.  The  air-pump. 

The  simplest  form  of  the  air-pump  is  represented 
in  figure  33,  in  which  EFF'E'  is  a  cylinder  furnished 
with  an  air-tight  piston  AB,  whose  play  is  EE' ;  S 
and  S',  two  valves  opening  upwards,  the  one  closing 
an  aperture  in  the  piston,  the  other  in  the  bottom  of 
the  cylinder;  and  PQRO  a  vessel  or  receiver  con- 
nected with  the  cylinder  by  a  pipe,  and  from  which 


260  INSTRUMENTS    AND    MACHINES. 

it  is  desired  to  extract  the  air.  It  is  obvious  that 
when  the  piston  is  raised  from  EF  to  E'  F' ,  the  valve 
S'  opens,  and  a  portion  of  the  air  in  the  receiver 
passes  into  the  cylinder ;  and  that  when  the  piston 
is  depressed  to  its  former  position,  this  same  mass  of 
air  escapes  by  the  valve  S  into  the  atmosphere. 
Each  stroke  of  the  piston  producing  a  similar  effect, 
the  air  in  the  receiver  may  thus  be  reduced  to  a  very 
low  degree  of  density.  As  ordinarily  constructed,  the 
air-pump  has  two  cylinders,  to  which  the  pistons  are 
so  adjusted,  that  when  one  of  them  is  ascending,  the 
other  is  descending. 

Let  V  denote  the  sum  of  the  volumes  of  the  re- 
ceiver and  pipe,  and  v'  the  volume  of  the  cylinder : 
also  let  d  denote  the  original  density  of  the  air  in  the 
receiver,  and  d' ,  d" ,  d'" ,  etc.  its  densities  after  the 
1st,  2d,  3d,  etc.  cylinder  of  air  is  withdrawn.  Then 
the  air  whose  volume  is  v  and  density  d,  being  first 
expanded  to  the  volume  v  +  v'  with  a  density  d' ,  we 
have 

v-\-v'     :     V     :   :     d     :     d'. 

Again,  the  air  whose  volume  is  v  and  density  d' ,  being 
expanded  to  the  volume  v-\-v'  with  a  density  d' ' ,  we 
have 

v-\-v'     :     V     :   :     d'     :     d" ; 

and  thus  we  have 

v  +  v'     :     V     :  :     d"     :     d"',   etc. 


Hence  we  have 


Mi^^- 


INSTRUMENTS   AND   MACHINES.  261 

v-\-v^  \v-\-v'l 

\v  +  v7  \v-\-v'J 

And  generally  after  the  ;2tli  cylinder  of  air  is  with- 
drawn, the  density  of  the  air  remaining  in  the  cylin- 
der will  be  expressed  by  (^^iy)"<^- 


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UNIVERSITY  OF  CALIFORNIA  LIBRARY 

,  Los  Angeles 
This  book  is  DUE  on  the  last  date  stamped  below. 


NOV   4      j|53 

FEB  ?        1954 


Form  L9-100m-9,'52(A3105)444 


2o  ^ 

Jl 


l^ 


STACK 


JUL72 


